src/HOL/Set.thy
author haftmann
Sat, 14 Mar 2009 12:50:29 +0100
changeset 30531 ab3d61baf66a
parent 30352 047f183c43b0
child 30596 140b22f22071
permissions -rw-r--r--
reverted to old version of Set.thy -- strange effects have to be traced first
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
12257
e3f7d6fb55d7 theory Inverse_Image converted and moved to Set;
wenzelm
parents: 12114
diff changeset
     2
    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     3
*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     4
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
     5
header {* Set theory for higher-order logic *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
     6
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15120
diff changeset
     7
theory Set
30304
d8e4cd2ac2a1 set operations Int, Un, INTER, UNION, Inter, Union, empty, UNIV are now proper qualified constants with authentic syntax
haftmann
parents: 29901
diff changeset
     8
imports Lattices
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15120
diff changeset
     9
begin
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    10
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    11
text {* A set in HOL is simply a predicate. *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    12
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    13
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    14
subsection {* Basic syntax *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    15
3947
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
    16
global
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
    17
26800
dcf1dfc915a7 - Now uses Orderings as parent theory
berghofe
parents: 26732
diff changeset
    18
types 'a set = "'a => bool"
3820
46b255e140dc fixed infix syntax;
wenzelm
parents: 3370
diff changeset
    19
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    20
consts
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    21
  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    22
  "op :"        :: "'a => 'a set => bool"                -- "membership"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    23
  insert        :: "'a => 'a set => 'a set"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    24
  Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    25
  Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    26
  Bex1          :: "'a set => ('a => bool) => bool"      -- "bounded unique existential quantifiers"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    27
  Pow           :: "'a set => 'a set set"                -- "powerset"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    28
  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
30304
d8e4cd2ac2a1 set operations Int, Un, INTER, UNION, Inter, Union, empty, UNIV are now proper qualified constants with authentic syntax
haftmann
parents: 29901
diff changeset
    29
d8e4cd2ac2a1 set operations Int, Un, INTER, UNION, Inter, Union, empty, UNIV are now proper qualified constants with authentic syntax
haftmann
parents: 29901
diff changeset
    30
local
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    31
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20380
diff changeset
    32
notation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    33
  "op :"  ("op :") and
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    34
  "op :"  ("(_/ : _)" [50, 51] 50)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
    35
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    36
abbreviation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    37
  "not_mem x A == ~ (x : A)" -- "non-membership"
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    38
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20380
diff changeset
    39
notation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    40
  not_mem  ("op ~:") and
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    41
  not_mem  ("(_/ ~: _)" [50, 51] 50)
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    42
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20380
diff changeset
    43
notation (xsymbols)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    44
  "op :"  ("op \<in>") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    45
  "op :"  ("(_/ \<in> _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    46
  not_mem  ("op \<notin>") and
30304
d8e4cd2ac2a1 set operations Int, Un, INTER, UNION, Inter, Union, empty, UNIV are now proper qualified constants with authentic syntax
haftmann
parents: 29901
diff changeset
    47
  not_mem  ("(_/ \<notin> _)" [50, 51] 50)
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    48
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20380
diff changeset
    49
notation (HTML output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    50
  "op :"  ("op \<in>") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    51
  "op :"  ("(_/ \<in> _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
    52
  not_mem  ("op \<notin>") and
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    53
  not_mem  ("(_/ \<notin> _)" [50, 51] 50)
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
    54
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    55
syntax
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    56
  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    57
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    58
translations
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    59
  "{x. P}"      == "Collect (%x. P)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    60
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    61
definition empty :: "'a set" ("{}") where
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    62
  "empty \<equiv> {x. False}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    63
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    64
definition UNIV :: "'a set" where
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    65
  "UNIV \<equiv> {x. True}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    66
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    67
syntax
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    68
  "@Finset"     :: "args => 'a set"                       ("{(_)}")
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    69
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    70
translations
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    71
  "{x, xs}"     == "insert x {xs}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    72
  "{x}"         == "insert x {}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    73
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    74
definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    75
  "A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    76
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    77
definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    78
  "A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    79
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    80
notation (xsymbols)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    81
  "Int"  (infixl "\<inter>" 70) and
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    82
  "Un"  (infixl "\<union>" 65)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    83
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    84
notation (HTML output)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    85
  "Int"  (infixl "\<inter>" 70) and
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    86
  "Un"  (infixl "\<union>" 65)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    87
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    88
syntax
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    89
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    90
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    91
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3EX! _:_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    92
  "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST _:_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    93
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    94
syntax (HOL)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    95
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    96
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    97
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3?! _:_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    98
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
    99
syntax (xsymbols)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   100
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   101
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   102
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   103
  "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   104
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   105
syntax (HTML output)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   106
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   107
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   108
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   109
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   110
translations
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   111
  "ALL x:A. P"  == "Ball A (%x. P)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   112
  "EX x:A. P"   == "Bex A (%x. P)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   113
  "EX! x:A. P"  == "Bex1 A (%x. P)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   114
  "LEAST x:A. P" => "LEAST x. x:A & P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   115
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   116
definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   117
  "INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   118
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   119
definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   120
  "UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   121
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   122
definition Inter :: "'a set set \<Rightarrow> 'a set" where
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   123
  "Inter S \<equiv> INTER S (\<lambda>x. x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   124
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   125
definition Union :: "'a set set \<Rightarrow> 'a set" where
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   126
  "Union S \<equiv> UNION S (\<lambda>x. x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   127
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   128
notation (xsymbols)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   129
  Inter  ("\<Inter>_" [90] 90) and
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   130
  Union  ("\<Union>_" [90] 90)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   131
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   132
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   133
subsection {* Additional concrete syntax *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   134
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   135
syntax
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   136
  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   137
  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   138
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   139
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   140
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   141
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   142
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   143
syntax (xsymbols)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   144
  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   145
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   146
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   147
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   148
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   149
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   150
syntax (latex output)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   151
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   152
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   153
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   154
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   155
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   156
translations
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   157
  "{x:A. P}"    => "{x. x:A & P}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   158
  "INT x y. B"  == "INT x. INT y. B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   159
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   160
  "INT x. B"    == "INT x:CONST UNIV. B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   161
  "INT x:A. B"  == "CONST INTER A (%x. B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   162
  "UN x y. B"   == "UN x. UN y. B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   163
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   164
  "UN x. B"     == "UN x:CONST UNIV. B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   165
  "UN x:A. B"   == "CONST UNION A (%x. B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   166
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   167
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   168
  Note the difference between ordinary xsymbol syntax of indexed
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   169
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   170
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   171
  former does not make the index expression a subscript of the
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   172
  union/intersection symbol because this leads to problems with nested
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   173
  subscripts in Proof General.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   174
*}
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
   175
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   176
abbreviation
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   177
  subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   178
  "subset \<equiv> less"
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   179
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   180
abbreviation
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   181
  subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   182
  "subset_eq \<equiv> less_eq"
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   183
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   184
notation (output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   185
  subset  ("op <") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   186
  subset  ("(_/ < _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   187
  subset_eq  ("op <=") and
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   188
  subset_eq  ("(_/ <= _)" [50, 51] 50)
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   189
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   190
notation (xsymbols)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   191
  subset  ("op \<subset>") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   192
  subset  ("(_/ \<subset> _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   193
  subset_eq  ("op \<subseteq>") and
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   194
  subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   195
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   196
notation (HTML output)
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   197
  subset  ("op \<subset>") and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   198
  subset  ("(_/ \<subset> _)" [50, 51] 50) and
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   199
  subset_eq  ("op \<subseteq>") and
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   200
  subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   201
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   202
abbreviation (input)
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   203
  supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   204
  "supset \<equiv> greater"
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   205
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21384
diff changeset
   206
abbreviation (input)
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   207
  supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   208
  "supset_eq \<equiv> greater_eq"
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   209
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   210
notation (xsymbols)
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   211
  supset  ("op \<supset>") and
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   212
  supset  ("(_/ \<supset> _)" [50, 51] 50) and
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   213
  supset_eq  ("op \<supseteq>") and
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   214
  supset_eq  ("(_/ \<supseteq> _)" [50, 51] 50)
21333
eb291029d6cd dropped LOrder dependency
haftmann
parents: 21316
diff changeset
   215
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   216
abbreviation
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   217
  range :: "('a => 'b) => 'b set" where -- "of function"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   218
  "range f == f ` UNIV"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   219
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   220
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   221
subsubsection "Bounded quantifiers"
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   222
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   223
syntax (output)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   224
  "_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
   225
  "_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
   226
  "_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
   227
  "_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
   228
  "_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
   229
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   230
syntax (xsymbols)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   231
  "_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
   232
  "_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
   233
  "_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
   234
  "_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
   235
  "_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
   236
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   237
syntax (HOL output)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   238
  "_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
   239
  "_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
   240
  "_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
   241
  "_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
   242
  "_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
   243
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   244
syntax (HTML output)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   245
  "_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
   246
  "_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
   247
  "_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
   248
  "_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
   249
  "_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
   250
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   251
translations
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   252
 "\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   253
 "\<exists>A\<subset>B. P"   =>  "EX A. A \<subset> B & P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   254
 "\<forall>A\<subseteq>B. P"   =>  "ALL A. A \<subseteq> B --> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   255
 "\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   256
 "\<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
   257
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   258
print_translation {*
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   259
let
22377
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   260
  val Type (set_type, _) = @{typ "'a set"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   261
  val All_binder = Syntax.binder_name @{const_syntax "All"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   262
  val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   263
  val impl = @{const_syntax "op -->"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   264
  val conj = @{const_syntax "op &"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   265
  val sbset = @{const_syntax "subset"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   266
  val sbset_eq = @{const_syntax "subset_eq"};
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   267
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   268
  val trans =
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   269
   [((All_binder, impl, sbset), "_setlessAll"),
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   270
    ((All_binder, impl, sbset_eq), "_setleAll"),
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   271
    ((Ex_binder, conj, sbset), "_setlessEx"),
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   272
    ((Ex_binder, conj, sbset_eq), "_setleEx")];
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   273
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   274
  fun mk v v' c n P =
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   275
    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
   276
    then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   277
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   278
  fun tr' q = (q,
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   279
    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
   280
         if T = (set_type) then case AList.lookup (op =) trans (q, c, d)
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   281
          of NONE => raise Match
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   282
           | SOME l => mk v v' l n P
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   283
         else raise Match
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   284
     | _ => raise Match);
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   285
in
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   286
  [tr' All_binder, tr' Ex_binder]
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   287
end
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   288
*}
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   289
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   290
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   291
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   292
  \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   293
  "{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
   294
  only translated if @{text "[0..n] subset bvs(e)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   295
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   296
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   297
parse_translation {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   298
  let
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   299
    val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
3947
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
   300
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   301
    fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   302
      | nvars _ = 1;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   303
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   304
    fun setcompr_tr [e, idts, b] =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   305
      let
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   306
        val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   307
        val P = Syntax.const "op &" $ eq $ b;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   308
        val exP = ex_tr [idts, P];
17784
5cbb52f2c478 Term.absdummy;
wenzelm
parents: 17715
diff changeset
   309
      in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   310
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   311
  in [("@SetCompr", setcompr_tr)] end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   312
*}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   313
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   314
(* To avoid eta-contraction of body: *)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   315
print_translation {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   316
let
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   317
  fun btr' syn [A, Abs abs] =
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   318
    let val (x, t) = atomic_abs_tr' abs
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   319
    in Syntax.const syn $ x $ A $ t end
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   320
in
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   321
[(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex"),
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   322
 (@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   323
end
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   324
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   325
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   326
print_translation {*
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   327
let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   328
  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
   329
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   330
  fun setcompr_tr' [Abs (abs as (_, _, P))] =
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   331
    let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   332
      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
   333
        | 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
   334
            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
   335
            ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
13764
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
   336
        | check _ = false
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   337
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   338
        fun tr' (_ $ abs) =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   339
          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   340
          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
   341
    in if check (P, 0) then tr' P
15535
nipkow
parents: 15524
diff changeset
   342
       else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
nipkow
parents: 15524
diff changeset
   343
                val M = Syntax.const "@Coll" $ x $ t
nipkow
parents: 15524
diff changeset
   344
            in case t of
nipkow
parents: 15524
diff changeset
   345
                 Const("op &",_)
nipkow
parents: 15524
diff changeset
   346
                   $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
nipkow
parents: 15524
diff changeset
   347
                   $ P =>
nipkow
parents: 15524
diff changeset
   348
                   if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
nipkow
parents: 15524
diff changeset
   349
               | _ => M
nipkow
parents: 15524
diff changeset
   350
            end
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   351
    end;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   352
  in [("Collect", setcompr_tr')] end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   353
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   354
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   355
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   356
subsection {* Rules and definitions *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   357
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   358
text {* Isomorphisms between predicates and sets. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   359
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   360
defs
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   361
  mem_def [code]: "x : S == S x"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   362
  Collect_def [code]: "Collect P == P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   363
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   364
defs
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   365
  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   366
  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   367
  Bex1_def:     "Bex1 A P       == EX! x. x:A & P(x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   368
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   369
instantiation "fun" :: (type, minus) minus
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   370
begin
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   371
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   372
definition
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   373
  fun_diff_def: "A - B = (%x. A x - B x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   374
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   375
instance ..
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   376
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   377
end
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   378
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   379
instantiation bool :: minus
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   380
begin
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   381
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   382
definition
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   383
  bool_diff_def: "A - B = (A & ~ B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   384
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   385
instance ..
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   386
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   387
end
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   388
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   389
instantiation "fun" :: (type, uminus) uminus
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   390
begin
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   391
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   392
definition
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   393
  fun_Compl_def: "- A = (%x. - A x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   394
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   395
instance ..
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   396
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   397
end
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   398
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   399
instantiation bool :: uminus
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   400
begin
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   401
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   402
definition
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   403
  bool_Compl_def: "- A = (~ A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   404
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   405
instance ..
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   406
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   407
end
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   408
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   409
defs
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   410
  Pow_def:      "Pow A          == {B. B <= A}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   411
  insert_def:   "insert a B     == {x. x=a} Un B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   412
  image_def:    "f`A            == {y. EX x:A. y = f(x)}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   413
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   414
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   415
subsection {* Lemmas and proof tool setup *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   416
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   417
subsubsection {* Relating predicates and sets *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   418
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   419
lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   420
  by (simp add: Collect_def mem_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   421
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   422
lemma Collect_mem_eq [simp]: "{x. x:A} = A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   423
  by (simp add: Collect_def mem_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   424
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   425
lemma CollectI: "P(a) ==> a : {x. P(x)}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   426
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   427
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   428
lemma CollectD: "a : {x. P(x)} ==> P(a)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   429
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   430
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   431
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   432
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   433
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   434
lemmas CollectE = CollectD [elim_format]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   435
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   436
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   437
subsubsection {* Bounded quantifiers *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   438
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   439
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   440
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   441
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   442
lemmas strip = impI allI ballI
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   443
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   444
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   445
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   446
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   447
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   448
  by (unfold Ball_def) blast
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   449
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   450
ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   451
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   452
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   453
  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   454
  @{prop "a:A"}; creates assumption @{prop "P a"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   455
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   456
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   457
ML {*
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   458
  fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   459
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   460
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   461
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   462
  Gives better instantiation for bound:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   463
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   464
26339
7825c83c9eff eliminated change_claset/simpset;
wenzelm
parents: 26150
diff changeset
   465
declaration {* fn _ =>
7825c83c9eff eliminated change_claset/simpset;
wenzelm
parents: 26150
diff changeset
   466
  Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   467
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   468
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   469
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   470
  -- {* Normally the best argument order: @{prop "P x"} constrains the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   471
    choice of @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   472
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   473
13113
5eb9be7b72a5 rev_bexI [intro?];
wenzelm
parents: 13103
diff changeset
   474
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   475
  -- {* The best argument order when there is only one @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   476
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   477
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   478
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
   479
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   480
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   481
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   482
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   483
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   484
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   485
  -- {* Trival rewrite rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   486
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   487
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   488
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   489
  -- {* Dual form for existentials. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   490
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   491
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   492
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   493
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   494
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   495
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   496
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   497
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   498
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
   499
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   500
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   501
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
   502
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   503
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   504
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
   505
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   506
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   507
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
   508
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   509
26480
544cef16045b replaced 'ML_setup' by 'ML';
wenzelm
parents: 26339
diff changeset
   510
ML {*
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   511
  local
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   512
    val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   513
    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
   514
    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   515
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   516
    val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   517
    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
   518
    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   519
  in
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   520
    val defBEX_regroup = Simplifier.simproc (the_context ())
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   521
      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   522
    val defBALL_regroup = Simplifier.simproc (the_context ())
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   523
      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   524
  end;
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   525
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   526
  Addsimprocs [defBALL_regroup, defBEX_regroup];
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   527
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   528
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   529
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   530
subsubsection {* Congruence rules *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   531
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   532
lemma ball_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   533
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   534
    (ALL x:A. P x) = (ALL x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   535
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   536
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   537
lemma strong_ball_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   538
  "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
   539
    (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
   540
  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
   541
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   542
lemma bex_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   543
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   544
    (EX x:A. P x) = (EX x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   545
  by (simp add: Bex_def cong: conj_cong)
1273
6960ec882bca added 8bit pragmas
regensbu
parents: 1068
diff changeset
   546
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   547
lemma strong_bex_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   548
  "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
   549
    (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
   550
  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
   551
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   552
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   553
subsubsection {* Subsets *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   554
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   555
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   556
  by (auto simp add: mem_def intro: predicate1I)
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   557
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   558
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   559
  \medskip Map the type @{text "'a set => anything"} to just @{typ
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   560
  'a}; for overloading constants whose first argument has type @{typ
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   561
  "'a set"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   562
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   563
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   564
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   565
  -- {* Rule in Modus Ponens style. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   566
  by (unfold mem_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   567
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   568
declare subsetD [intro?] -- FIXME
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   569
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   570
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   571
  -- {* The same, with reversed premises for use with @{text erule} --
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   572
      cf @{text rev_mp}. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   573
  by (rule subsetD)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   574
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   575
declare rev_subsetD [intro?] -- FIXME
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   576
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   577
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   578
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   579
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   580
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   581
ML {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   582
  fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   583
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   584
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   585
lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   586
  -- {* Classical elimination rule. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   587
  by (unfold mem_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   588
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   589
lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   590
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   591
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   592
  \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   593
  creates the assumption @{prop "c \<in> B"}.
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   594
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   595
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   596
ML {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   597
  fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   598
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   599
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   600
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   601
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   602
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   603
lemma subset_refl [simp,atp]: "A \<subseteq> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   604
  by fast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   605
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   606
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   607
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   608
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   609
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   610
subsubsection {* Equality *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   611
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   612
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   613
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   614
   apply (rule Collect_mem_eq)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   615
  apply (rule Collect_mem_eq)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   616
  done
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   617
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   618
(* Due to Brian Huffman *)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   619
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   620
by(auto intro:set_ext)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   621
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   622
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   623
  -- {* Anti-symmetry of the subset relation. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   624
  by (iprover intro: set_ext subsetD)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   625
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   626
lemmas equalityI [intro!] = subset_antisym
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   627
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   628
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   629
  \medskip Equality rules from ZF set theory -- are they appropriate
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   630
  here?
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   631
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   632
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   633
lemma equalityD1: "A = B ==> A \<subseteq> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   634
  by (simp add: subset_refl)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   635
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   636
lemma equalityD2: "A = B ==> B \<subseteq> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   637
  by (simp add: subset_refl)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   638
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   639
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   640
  \medskip Be careful when adding this to the claset as @{text
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   641
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   642
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   643
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   644
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   645
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   646
  by (simp add: subset_refl)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   647
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   648
lemma equalityCE [elim]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   649
    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   650
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   651
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   652
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   653
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   654
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   655
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   656
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   657
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   658
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   659
subsubsection {* The universal set -- UNIV *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   660
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   661
lemma UNIV_I [simp]: "x : UNIV"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   662
  by (simp add: UNIV_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   663
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   664
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   665
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   666
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   667
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   668
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   669
lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   670
  by (rule subsetI) (rule UNIV_I)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   671
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   672
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   673
  \medskip Eta-contracting these two rules (to remove @{text P})
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   674
  causes them to be ignored because of their interaction with
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   675
  congruence rules.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   676
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   677
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   678
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   679
  by (simp add: Ball_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   680
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   681
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   682
  by (simp add: Bex_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   683
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   684
lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   685
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   686
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   687
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   688
subsubsection {* The empty set *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   689
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   690
lemma empty_iff [simp]: "(c : {}) = False"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   691
  by (simp add: empty_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   692
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   693
lemma emptyE [elim!]: "a : {} ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   694
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   695
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   696
lemma empty_subsetI [iff]: "{} \<subseteq> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   697
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   698
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   699
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   700
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   701
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   702
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   703
lemma equals0D: "A = {} ==> a \<notin> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   704
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   705
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   706
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   707
lemma ball_empty [simp]: "Ball {} P = True"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   708
  by (simp add: Ball_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   709
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   710
lemma bex_empty [simp]: "Bex {} P = False"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   711
  by (simp add: Bex_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   712
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   713
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   714
  by (blast elim: equalityE)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   715
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   716
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   717
subsubsection {* The Powerset operator -- Pow *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   718
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   719
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   720
  by (simp add: Pow_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   721
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   722
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   723
  by (simp add: Pow_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   724
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   725
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   726
  by (simp add: Pow_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   727
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   728
lemma Pow_bottom: "{} \<in> Pow B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   729
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   730
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   731
lemma Pow_top: "A \<in> Pow A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   732
  by (simp add: subset_refl)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   733
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   734
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   735
subsubsection {* Set complement *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   736
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   737
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   738
  by (simp add: mem_def fun_Compl_def bool_Compl_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   739
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   740
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   741
  by (unfold mem_def fun_Compl_def bool_Compl_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   742
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   743
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   744
  \medskip This form, with negated conclusion, works well with the
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   745
  Classical prover.  Negated assumptions behave like formulae on the
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   746
  right side of the notional turnstile ... *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   747
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   748
lemma ComplD [dest!]: "c : -A ==> c~:A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   749
  by (simp add: mem_def fun_Compl_def bool_Compl_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   750
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   751
lemmas ComplE = ComplD [elim_format]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   752
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   753
lemma Compl_eq: "- A = {x. ~ x : A}" by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   754
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   755
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   756
subsubsection {* Binary union -- Un *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   757
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   758
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   759
  by (unfold Un_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   760
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   761
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   762
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   763
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   764
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   765
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   766
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   767
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   768
  \medskip Classical introduction rule: no commitment to @{prop A} vs
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   769
  @{prop B}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   770
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   771
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   772
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   773
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   774
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   775
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   776
  by (unfold Un_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   777
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   778
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   779
subsubsection {* Binary intersection -- Int *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   780
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   781
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   782
  by (unfold Int_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   783
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   784
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   785
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   786
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   787
lemma IntD1: "c : A Int B ==> c:A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   788
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   789
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   790
lemma IntD2: "c : A Int B ==> c:B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   791
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   792
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   793
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   794
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   795
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   796
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   797
subsubsection {* Set difference *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   798
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   799
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   800
  by (simp add: mem_def fun_diff_def bool_diff_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   801
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   802
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   803
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   804
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   805
lemma DiffD1: "c : A - B ==> c : A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   806
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   807
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   808
lemma DiffD2: "c : A - B ==> c : B ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   809
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   810
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   811
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   812
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   813
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   814
lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   815
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   816
lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   817
by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   818
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   819
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   820
subsubsection {* Augmenting a set -- insert *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   821
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   822
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   823
  by (unfold insert_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   824
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   825
lemma insertI1: "a : insert a B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   826
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   827
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   828
lemma insertI2: "a : B ==> a : insert b B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   829
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   830
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   831
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   832
  by (unfold insert_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   833
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   834
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   835
  -- {* Classical introduction rule. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   836
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   837
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   838
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   839
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   840
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   841
lemma set_insert:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   842
  assumes "x \<in> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   843
  obtains B where "A = insert x B" and "x \<notin> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   844
proof
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   845
  from assms show "A = insert x (A - {x})" by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   846
next
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   847
  show "x \<notin> A - {x}" by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   848
qed
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   849
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   850
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   851
by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   852
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   853
subsubsection {* Singletons, using insert *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   854
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   855
lemma singletonI [intro!,noatp]: "a : {a}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   856
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   857
  by (rule insertI1)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   858
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   859
lemma singletonD [dest!,noatp]: "b : {a} ==> b = a"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   860
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   861
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   862
lemmas singletonE = singletonD [elim_format]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   863
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   864
lemma singleton_iff: "(b : {a}) = (b = a)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   865
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   866
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   867
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   868
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   869
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   870
lemma singleton_insert_inj_eq [iff,noatp]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   871
     "({b} = insert a A) = (a = b & A \<subseteq> {b})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   872
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   873
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   874
lemma singleton_insert_inj_eq' [iff,noatp]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   875
     "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   876
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   877
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   878
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   879
  by fast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   880
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   881
lemma singleton_conv [simp]: "{x. x = a} = {a}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   882
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   883
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   884
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   885
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   886
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   887
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   888
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   889
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   890
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   891
  by (blast elim: equalityE)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   892
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   893
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   894
subsubsection {* Unions of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   895
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   896
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   897
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   898
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   899
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
   900
declare UNION_def [noatp]
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
   901
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   902
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
   903
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   904
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   905
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   906
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   907
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   908
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   909
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   910
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
   911
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   912
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   913
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   914
    "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
   915
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   916
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   917
lemma strong_UN_cong:
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   918
    "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   919
  by (simp add: UNION_def simp_implies_def)
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   920
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   921
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   922
subsubsection {* Intersections of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   923
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   924
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   925
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   926
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
   927
  by (unfold INTER_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   928
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   929
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
   930
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   931
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   932
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   933
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   934
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   935
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
   936
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   937
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   938
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   939
lemma INT_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   940
    "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
   941
  by (simp add: INTER_def)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   942
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   943
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   944
subsubsection {* Union *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   945
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
   946
lemma Union_iff [simp,noatp]: "(A : Union C) = (EX X:C. A:X)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   947
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   948
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   949
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   950
  -- {* The order of the premises presupposes that @{term C} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   951
    @{term A} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   952
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   953
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   954
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   955
  by (unfold Union_def) blast
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
subsubsection {* Inter *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   959
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
   960
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   961
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   962
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   963
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   964
  by (simp add: Inter_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   965
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   966
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   967
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   968
  contains @{term A} as an element, but @{prop "A:X"} can hold when
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   969
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   970
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   971
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   972
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   973
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   974
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   975
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   976
  -- {* ``Classical'' elimination rule -- does not require proving
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   977
    @{prop "X:C"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   978
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   979
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   980
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   981
  \medskip Image of a set under a function.  Frequently @{term b} does
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   982
  not have the syntactic form of @{term "f x"}.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   983
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   984
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   985
declare image_def [noatp]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   986
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   987
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   988
  by (unfold image_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   989
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   990
lemma imageI: "x : A ==> f x : f ` A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   991
  by (rule image_eqI) (rule refl)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   992
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   993
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   994
  -- {* This version's more effective when we already have the
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   995
    required @{term x}. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   996
  by (unfold image_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   997
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   998
lemma imageE [elim!]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   999
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1000
  -- {* The eta-expansion gives variable-name preservation. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1001
  by (unfold image_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1002
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1003
lemma image_Un: "f`(A Un B) = f`A Un f`B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1004
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1005
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1006
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1007
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1008
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1009
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1010
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1011
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1012
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1013
  -- {* This rewrite rule would confuse users if made default. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1014
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1015
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1016
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1017
  apply safe
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1018
   prefer 2 apply fast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1019
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1020
  done
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1021
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1022
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1023
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1024
    @{text hypsubst}, but breaks too many existing proofs. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1025
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1026
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1027
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1028
  \medskip Range of a function -- just a translation for image!
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1029
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1030
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1031
lemma range_eqI: "b = f x ==> b \<in> range f"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1032
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1033
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1034
lemma rangeI: "f x \<in> range f"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1035
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1036
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1037
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1038
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1039
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1040
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1041
subsubsection {* Set reasoning tools *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1042
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1043
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1044
  Rewrite rules for boolean case-splitting: faster than @{text
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1045
  "split_if [split]"}.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1046
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1047
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1048
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1049
  by (rule split_if)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1050
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1051
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1052
  by (rule split_if)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1053
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1054
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1055
  Split ifs on either side of the membership relation.  Not for @{text
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1056
  "[simp]"} -- can cause goals to blow up!
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1057
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1058
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1059
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1060
  by (rule split_if)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1061
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1062
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1063
  by (rule split_if [where P="%S. a : S"])
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1064
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1065
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1066
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1067
lemmas mem_simps =
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1068
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1069
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1070
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1071
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1072
(*Would like to add these, but the existing code only searches for the
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1073
  outer-level constant, which in this case is just "op :"; we instead need
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1074
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1075
  apply, then the formula should be kept.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1076
  [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1077
   ("Int", [IntD1,IntD2]),
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1078
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1079
 *)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1080
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1081
ML {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1082
  val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1083
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1084
declaration {* fn _ =>
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1085
  Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1086
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1087
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1088
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1089
subsubsection {* The ``proper subset'' relation *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1090
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1091
lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1092
  by (unfold less_le) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1093
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1094
lemma psubsetE [elim!,noatp]: 
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1095
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1096
  by (unfold less_le) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1097
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1098
lemma psubset_insert_iff:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1099
  "(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)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1100
  by (auto simp add: less_le subset_insert_iff)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1101
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1102
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1103
  by (simp only: less_le)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1104
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1105
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1106
  by (simp add: psubset_eq)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1107
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1108
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1109
apply (unfold less_le)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1110
apply (auto dest: subset_antisym)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1111
done
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1112
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1113
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1114
apply (unfold less_le)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1115
apply (auto dest: subsetD)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1116
done
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1117
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1118
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1119
  by (auto simp add: psubset_eq)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1120
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1121
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1122
  by (auto simp add: psubset_eq)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1123
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1124
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1125
  by (unfold less_le) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1126
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1127
lemma atomize_ball:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1128
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1129
  by (simp only: Ball_def atomize_all atomize_imp)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1130
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1131
lemmas [symmetric, rulify] = atomize_ball
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1132
  and [symmetric, defn] = atomize_ball
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1133
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1134
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1135
subsection {* Further set-theory lemmas *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1136
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1137
subsubsection {* Derived rules involving subsets. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1138
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1139
text {* @{text insert}. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1140
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1141
lemma subset_insertI: "B \<subseteq> insert a B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1142
  by (rule subsetI) (erule insertI2)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1143
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1144
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1145
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1146
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1147
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1148
  by blast
12897
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
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1151
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
  1152
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1153
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1154
  by (iprover intro: subsetI UnionI)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1155
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1156
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
  1157
  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
  1158
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
text {* \medskip General union. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1161
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1162
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
  1163
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1164
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1165
lemma 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
  1166
  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
  1167
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
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
  1170
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1171
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
  1172
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1173
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1174
lemma Inter_subset:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1175
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1176
  by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1177
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1178
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
  1179
  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
  1180
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1181
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
  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 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
  1185
  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
  1186
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1187
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1188
text {* \medskip Finite Union -- the least upper bound of two sets. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1189
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1190
lemma Un_upper1: "A \<subseteq> A \<union> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1191
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1192
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1193
lemma Un_upper2: "B \<subseteq> A \<union> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1194
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1195
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1196
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1197
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1198
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1199
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1200
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1201
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1202
lemma Int_lower1: "A \<inter> B \<subseteq> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1203
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1204
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1205
lemma Int_lower2: "A \<inter> B \<subseteq> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1206
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1207
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1208
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1209
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1210
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1211
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1212
text {* \medskip Set difference. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1213
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1214
lemma Diff_subset: "A - B \<subseteq> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1215
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1216
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1217
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1218
by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1219
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1220
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1221
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1222
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1223
text {* @{text "{}"}. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1224
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1225
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1226
  -- {* supersedes @{text "Collect_False_empty"} *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1227
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1228
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1229
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1230
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1231
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1232
lemma not_psubset_empty [iff]: "\<not> (A < {})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1233
  by (unfold less_le) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1234
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1235
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1236
by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1237
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1238
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1239
by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1240
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1241
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1242
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1243
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1244
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1245
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1246
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1247
lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1248
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1249
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1250
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1251
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1252
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1253
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
  1254
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1255
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1256
lemma 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
  1257
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1258
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
  1259
lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1260
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1261
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
  1262
lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1263
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1264
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1265
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1266
text {* \medskip @{text insert}. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1267
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1268
lemma insert_is_Un: "insert a A = {a} Un A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1269
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1270
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1271
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1272
lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1273
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1274
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1275
lemmas empty_not_insert = insert_not_empty [symmetric, standard]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1276
declare empty_not_insert [simp]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1277
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1278
lemma insert_absorb: "a \<in> A ==> insert a A = A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1279
  -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1280
  -- {* with \emph{quadratic} running time *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1281
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1282
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1283
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1284
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1285
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1286
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1287
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1288
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1289
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1290
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1291
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1292
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1293
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1294
  apply (rule_tac x = "A - {a}" in exI, blast)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1295
  done
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1296
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1297
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1298
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1299
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1300
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1301
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1302
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1303
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
  1304
  by blast
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1305
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1306
lemma insert_disjoint [simp,noatp]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1307
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1308
 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1309
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1310
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1311
lemma disjoint_insert [simp,noatp]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1312
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1313
 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1314
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1315
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1316
text {* \medskip @{text image}. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1317
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1318
lemma image_empty [simp]: "f`{} = {}"