src/HOL/Complete_Lattice.thy
author huffman
Sat, 06 Mar 2010 08:08:30 -0800
changeset 35629 57f1a5e93b6b
parent 35115 446c5063e4fd
child 35828 46cfc4b8112e
permissions -rw-r--r--
add some lemmas about complete lattices
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
     1
(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
     2
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
     3
header {* Complete lattices, with special focus on sets *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
     4
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
     5
theory Complete_Lattice
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
     6
imports Set
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
     7
begin
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
     8
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
     9
notation
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    10
  less_eq (infix "\<sqsubseteq>" 50) and
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    11
  less (infix "\<sqsubset>" 50) and
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    12
  inf (infixl "\<sqinter>" 70) and
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    13
  sup (infixl "\<squnion>" 65) and
32678
de1f7d4da21a added dual for complete lattice
haftmann
parents: 32642
diff changeset
    14
  top ("\<top>") and
de1f7d4da21a added dual for complete lattice
haftmann
parents: 32642
diff changeset
    15
  bot ("\<bottom>")
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    16
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
    17
32879
7f5ce7af45fd added syntactic Inf and Sup
haftmann
parents: 32678
diff changeset
    18
subsection {* Syntactic infimum and supremum operations *}
7f5ce7af45fd added syntactic Inf and Sup
haftmann
parents: 32678
diff changeset
    19
7f5ce7af45fd added syntactic Inf and Sup
haftmann
parents: 32678
diff changeset
    20
class Inf =
7f5ce7af45fd added syntactic Inf and Sup
haftmann
parents: 32678
diff changeset
    21
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
7f5ce7af45fd added syntactic Inf and Sup
haftmann
parents: 32678
diff changeset
    22
7f5ce7af45fd added syntactic Inf and Sup
haftmann
parents: 32678
diff changeset
    23
class Sup =
7f5ce7af45fd added syntactic Inf and Sup
haftmann
parents: 32678
diff changeset
    24
  fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
7f5ce7af45fd added syntactic Inf and Sup
haftmann
parents: 32678
diff changeset
    25
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
    26
subsection {* Abstract complete lattices *}
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
    27
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    28
class complete_lattice = bounded_lattice + Inf + Sup +
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    29
  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    30
     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    31
  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    32
     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    33
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    34
32678
de1f7d4da21a added dual for complete lattice
haftmann
parents: 32642
diff changeset
    35
lemma dual_complete_lattice:
32879
7f5ce7af45fd added syntactic Inf and Sup
haftmann
parents: 32678
diff changeset
    36
  "complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    37
  by (auto intro!: complete_lattice.intro dual_bounded_lattice)
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    38
    (unfold_locales, (fact bot_least top_greatest
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    39
        Sup_upper Sup_least Inf_lower Inf_greatest)+)
32678
de1f7d4da21a added dual for complete lattice
haftmann
parents: 32642
diff changeset
    40
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    41
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    42
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    43
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    44
lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    45
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    46
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    47
lemma Inf_empty:
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    48
  "\<Sqinter>{} = \<top>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    49
  by (auto intro: antisym Inf_greatest)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    50
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    51
lemma Sup_empty:
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    52
  "\<Squnion>{} = \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    53
  by (auto intro: antisym Sup_least)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    54
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    55
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    56
  by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    57
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    58
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    59
  by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    60
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    61
lemma Inf_singleton [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    62
  "\<Sqinter>{a} = a"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    63
  by (auto intro: antisym Inf_lower Inf_greatest)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    64
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    65
lemma Sup_singleton [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    66
  "\<Squnion>{a} = a"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    67
  by (auto intro: antisym Sup_upper Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    68
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    69
lemma Inf_binary:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    70
  "\<Sqinter>{a, b} = a \<sqinter> b"
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    71
  by (simp add: Inf_empty Inf_insert)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    72
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    73
lemma Sup_binary:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    74
  "\<Squnion>{a, b} = a \<squnion> b"
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    75
  by (simp add: Sup_empty Sup_insert)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    76
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    77
lemma Inf_UNIV:
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    78
  "\<Sqinter>UNIV = bot"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    79
  by (simp add: Sup_Inf Sup_empty [symmetric])
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    80
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    81
lemma Sup_UNIV:
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    82
  "\<Squnion>UNIV = top"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
    83
  by (simp add: Inf_Sup Inf_empty [symmetric])
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    84
35629
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
    85
lemma Sup_le_iff: "Sup A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
    86
  by (auto intro: Sup_least dest: Sup_upper)
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
    87
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
    88
lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
    89
  by (auto intro: Inf_greatest dest: Inf_lower)
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
    90
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    91
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
    92
  "SUPR A f = \<Squnion> (f ` A)"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    93
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    94
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
    95
  "INFI A f = \<Sqinter> (f ` A)"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    96
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    97
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    98
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    99
syntax
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   100
  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   101
  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   102
  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   103
  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   104
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   105
translations
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   106
  "SUP x y. B"   == "SUP x. SUP y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   107
  "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   108
  "SUP x. B"     == "SUP x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   109
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   110
  "INF x y. B"   == "INF x. INF y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   111
  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   112
  "INF x. B"     == "INF x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   113
  "INF x:A. B"   == "CONST INFI A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   114
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   115
print_translation {*
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   116
  [Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"},
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   117
    Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}]
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   118
*} -- {* to avoid eta-contraction of body *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   119
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   120
context complete_lattice
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   121
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   122
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
   123
lemma le_SUPI: "i : A \<Longrightarrow> M i \<sqsubseteq> (SUP i:A. M i)"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   124
  by (auto simp add: SUPR_def intro: Sup_upper)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   125
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
   126
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> u"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   127
  by (auto simp add: SUPR_def intro: Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   128
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
   129
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> M i"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   130
  by (auto simp add: INFI_def intro: Inf_lower)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   131
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32879
diff changeset
   132
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   133
  by (auto simp add: INFI_def intro: Inf_greatest)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   134
35629
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   135
lemma SUP_le_iff: "(SUP i:A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   136
  unfolding SUPR_def by (auto simp add: Sup_le_iff)
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   137
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   138
lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   139
  unfolding INFI_def by (auto simp add: le_Inf_iff)
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   140
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   141
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   142
  by (auto intro: antisym SUP_leI le_SUPI)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   143
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   144
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   145
  by (auto intro: antisym INF_leI le_INFI)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   146
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   147
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   148
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   149
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   150
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   151
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   152
instantiation bool :: complete_lattice
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   153
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   154
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   155
definition
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   156
  Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   157
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   158
definition
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   159
  Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   160
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   161
instance proof
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   162
qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   163
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   164
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   165
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   166
lemma Inf_empty_bool [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   167
  "\<Sqinter>{}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   168
  unfolding Inf_bool_def by auto
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   169
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   170
lemma not_Sup_empty_bool [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   171
  "\<not> \<Squnion>{}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   172
  unfolding Sup_bool_def by auto
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   173
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   174
lemma INFI_bool_eq:
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   175
  "INFI = Ball"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   176
proof (rule ext)+
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   177
  fix A :: "'a set"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   178
  fix P :: "'a \<Rightarrow> bool"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   179
  show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   180
    by (auto simp add: Ball_def INFI_def Inf_bool_def)
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   181
qed
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   182
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   183
lemma SUPR_bool_eq:
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   184
  "SUPR = Bex"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   185
proof (rule ext)+
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   186
  fix A :: "'a set"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   187
  fix P :: "'a \<Rightarrow> bool"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   188
  show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   189
    by (auto simp add: Bex_def SUPR_def Sup_bool_def)
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   190
qed
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   191
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   192
instantiation "fun" :: (type, complete_lattice) complete_lattice
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   193
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   194
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   195
definition
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   196
  Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   197
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   198
definition
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   199
  Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   200
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   201
instance proof
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   202
qed (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   203
  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   204
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   205
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   206
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   207
lemma Inf_empty_fun:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   208
  "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   209
  by (simp add: Inf_fun_def)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   210
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   211
lemma Sup_empty_fun:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   212
  "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   213
  by (simp add: Sup_fun_def)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   214
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   215
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   216
subsection {* Union *}
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   217
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   218
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   219
  "Union S \<equiv> \<Squnion>S"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   220
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   221
notation (xsymbols)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   222
  Union  ("\<Union>_" [90] 90)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   223
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   224
lemma Union_eq:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   225
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   226
proof (rule set_ext)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   227
  fix x
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   228
  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   229
    by auto
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   230
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   231
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   232
qed
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   233
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   234
lemma Union_iff [simp, noatp]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   235
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   236
  by (unfold Union_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   237
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   238
lemma UnionI [intro]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   239
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   240
  -- {* The order of the premises presupposes that @{term C} is rigid;
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   241
    @{term A} may be flexible. *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   242
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   243
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   244
lemma UnionE [elim!]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   245
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   246
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   247
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   248
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   249
  by (iprover intro: subsetI UnionI)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   250
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   251
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   252
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   253
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   254
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   255
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   256
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   257
lemma Union_empty [simp]: "Union({}) = {}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   258
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   259
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   260
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   261
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   262
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   263
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   264
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   265
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   266
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   267
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   268
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   269
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   270
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   271
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   272
lemma Union_empty_conv [simp,noatp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   273
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   274
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   275
lemma empty_Union_conv [simp,noatp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   276
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   277
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   278
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   279
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   280
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   281
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   282
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   283
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   284
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   285
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   286
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   287
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   288
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   289
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   290
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   291
subsection {* Unions of families *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   292
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   293
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   294
  "UNION \<equiv> SUPR"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   295
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   296
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   297
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   298
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   299
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   300
syntax (xsymbols)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   301
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   302
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   303
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   304
syntax (latex output)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   305
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   306
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   307
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   308
translations
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   309
  "UN x y. B"   == "UN x. UN y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   310
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   311
  "UN x. B"     == "UN x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   312
  "UN x:A. B"   == "CONST UNION A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   313
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   314
text {*
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   315
  Note the difference between ordinary xsymbol syntax of indexed
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   316
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   317
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   318
  former does not make the index expression a subscript of the
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   319
  union/intersection symbol because this leads to problems with nested
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   320
  subscripts in Proof General.
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   321
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   322
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   323
print_translation {*
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   324
  [Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   325
*} -- {* to avoid eta-contraction of body *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   326
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   327
lemma UNION_eq_Union_image:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   328
  "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   329
  by (fact SUPR_def)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   330
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   331
lemma Union_def:
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   332
  "\<Union>S = (\<Union>x\<in>S. x)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   333
  by (simp add: UNION_eq_Union_image image_def)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   334
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   335
lemma UNION_def [noatp]:
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   336
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   337
  by (auto simp add: UNION_eq_Union_image Union_eq)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   338
  
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   339
lemma Union_image_eq [simp]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   340
  "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   341
  by (rule sym) (fact UNION_eq_Union_image)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   342
  
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   343
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
   344
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   345
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   346
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   347
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   348
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   349
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   350
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   351
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
   352
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   353
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   354
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   355
    "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
   356
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   357
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   358
lemma strong_UN_cong:
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   359
    "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
   360
  by (simp add: UNION_def simp_implies_def)
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   361
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   362
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   363
  by blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   364
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   365
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   366
  by (fact le_SUPI)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   367
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   368
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   369
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   370
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   371
lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   372
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   373
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   374
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   375
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   376
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   377
lemma UN_empty [simp,noatp]: "(\<Union>x\<in>{}. B x) = {}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   378
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   379
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   380
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   381
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   382
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   383
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   384
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   385
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   386
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   387
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   388
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   389
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   390
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   391
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   392
lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   393
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   394
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   395
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   396
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   397
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   398
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
35629
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   399
  by (fact SUP_le_iff)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   400
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   401
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   402
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   403
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   404
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   405
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   406
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   407
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   408
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   409
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   410
lemma UNION_empty_conv[simp]:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   411
  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   412
  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   413
by blast+
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   414
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   415
lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   416
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   417
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   418
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   419
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   420
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   421
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   422
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   423
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   424
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   425
  by (auto simp add: split_if_mem2)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   426
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   427
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   428
  by (auto intro: bool_contrapos)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   429
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   430
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   431
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   432
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   433
lemma UN_mono:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   434
  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   435
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   436
  by (blast dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   437
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   438
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   439
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   440
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   441
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   442
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   443
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   444
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   445
  -- {* NOT suitable for rewriting *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   446
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   447
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   448
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   449
by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   450
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   451
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   452
subsection {* Inter *}
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   453
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   454
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   455
  "Inter S \<equiv> \<Sqinter>S"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   456
  
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   457
notation (xsymbols)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   458
  Inter  ("\<Inter>_" [90] 90)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   459
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   460
lemma Inter_eq [code del]:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   461
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   462
proof (rule set_ext)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   463
  fix x
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   464
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   465
    by auto
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   466
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   467
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   468
qed
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   469
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   470
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   471
  by (unfold Inter_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   472
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   473
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   474
  by (simp add: Inter_eq)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   475
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   476
text {*
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   477
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   478
  contains @{term A} as an element, but @{prop "A:X"} can hold when
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   479
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   480
*}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   481
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   482
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   483
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   484
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   485
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   486
  -- {* ``Classical'' elimination rule -- does not require proving
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   487
    @{prop "X:C"}. *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   488
  by (unfold Inter_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   489
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   490
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   491
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   492
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   493
lemma Inter_subset:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   494
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   495
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   496
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   497
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   498
  by (iprover intro: InterI subsetI dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   499
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   500
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   501
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   502
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   503
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   504
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   505
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   506
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   507
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   508
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   509
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   510
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   511
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   512
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   513
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   514
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   515
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   516
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   517
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   518
lemma Inter_UNIV_conv [simp,noatp]:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   519
  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   520
  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   521
  by blast+
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   522
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   523
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   524
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   525
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   526
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   527
subsection {* Intersections of families *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   528
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   529
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   530
  "INTER \<equiv> INFI"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   531
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   532
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   533
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   534
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   535
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   536
syntax (xsymbols)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   537
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   538
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   539
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   540
syntax (latex output)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   541
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   542
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   543
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   544
translations
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   545
  "INT x y. B"  == "INT x. INT y. B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   546
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   547
  "INT x. B"    == "INT x:CONST UNIV. B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   548
  "INT x:A. B"  == "CONST INTER A (%x. B)"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   549
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   550
print_translation {*
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   551
  [Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   552
*} -- {* to avoid eta-contraction of body *}
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   553
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   554
lemma INTER_eq_Inter_image:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   555
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   556
  by (fact INFI_def)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   557
  
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   558
lemma Inter_def:
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   559
  "\<Inter>S = (\<Inter>x\<in>S. x)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   560
  by (simp add: INTER_eq_Inter_image image_def)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   561
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   562
lemma INTER_def:
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   563
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   564
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   565
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   566
lemma Inter_image_eq [simp]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   567
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   568
  by (rule sym) (fact INTER_eq_Inter_image)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   569
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   570
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
   571
  by (unfold INTER_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   572
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   573
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
   574
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   575
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   576
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   577
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   578
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   579
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
   580
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   581
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   582
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   583
lemma INT_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   584
    "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
   585
  by (simp add: INTER_def)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   586
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   587
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   588
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   589
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   590
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   591
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   592
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   593
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   594
  by (fact INF_leI)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   595
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   596
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   597
  by (fact le_INFI)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   598
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   599
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   600
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   601
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   602
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   603
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   604
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   605
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
35629
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   606
  by (fact le_INF_iff)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   607
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   608
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   609
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   610
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   611
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   612
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   613
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   614
lemma INT_insert_distrib:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   615
    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   616
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   617
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   618
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   619
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   620
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   621
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   622
  -- {* Look: it has an \emph{existential} quantifier *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   623
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   624
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   625
lemma INTER_UNIV_conv[simp]:
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
   626
 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
   627
 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
   628
by blast+
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   629
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   630
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   631
  by (auto intro: bool_induct)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   632
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   633
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   634
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   635
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   636
lemma INT_anti_mono:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   637
  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   638
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   639
  -- {* The last inclusion is POSITIVE! *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   640
  by (blast dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   641
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   642
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   643
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   644
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   645
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   646
subsection {* Distributive laws *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   647
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   648
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   649
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   650
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   651
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   652
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   653
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   654
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   655
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   656
  -- {* Union of a family of unions *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   657
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   658
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   659
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   660
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   661
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   662
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   663
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   664
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   665
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   666
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   667
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   668
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   669
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   670
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   671
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   672
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   673
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   674
  -- {* Halmos, Naive Set Theory, page 35. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   675
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   676
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   677
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   678
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   679
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   680
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   681
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   682
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   683
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   684
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   685
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   686
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   687
subsection {* Complement *}
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   688
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   689
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   690
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   691
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   692
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   693
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   694
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   695
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   696
subsection {* Miniscoping and maxiscoping *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   697
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   698
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   699
           and Intersections. *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   700
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   701
lemma UN_simps [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   702
  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   703
  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   704
  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   705
  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   706
  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   707
  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   708
  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   709
  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   710
  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   711
  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   712
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   713
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   714
lemma INT_simps [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   715
  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   716
  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   717
  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   718
  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   719
  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   720
  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   721
  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   722
  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   723
  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   724
  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   725
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   726
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
   727
lemma ball_simps [simp,noatp]:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   728
  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   729
  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   730
  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   731
  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   732
  "!!P. (ALL x:{}. P x) = True"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   733
  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   734
  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   735
  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   736
  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   737
  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   738
  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   739
  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   740
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   741
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
   742
lemma bex_simps [simp,noatp]:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   743
  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   744
  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   745
  "!!P. (EX x:{}. P x) = False"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   746
  "!!P. (EX x:UNIV. P x) = (EX x. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   747
  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   748
  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   749
  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   750
  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   751
  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   752
  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   753
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   754
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   755
lemma ball_conj_distrib:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   756
  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   757
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   758
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   759
lemma bex_disj_distrib:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   760
  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   761
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   762
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   763
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   764
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   765
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   766
lemma UN_extend_simps:
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   767
  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   768
  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   769
  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   770
  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   771
  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   772
  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   773
  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   774
  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   775
  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   776
  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   777
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   778
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   779
lemma INT_extend_simps:
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   780
  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   781
  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   782
  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   783
  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   784
  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   785
  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   786
  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   787
  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   788
  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   789
  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   790
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   791
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   792
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   793
no_notation
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   794
  less_eq  (infix "\<sqsubseteq>" 50) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   795
  less (infix "\<sqsubset>" 50) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   796
  inf  (infixl "\<sqinter>" 70) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   797
  sup  (infixl "\<squnion>" 65) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   798
  Inf  ("\<Sqinter>_" [900] 900) and
32678
de1f7d4da21a added dual for complete lattice
haftmann
parents: 32642
diff changeset
   799
  Sup  ("\<Squnion>_" [900] 900) and
de1f7d4da21a added dual for complete lattice
haftmann
parents: 32642
diff changeset
   800
  top ("\<top>") and
de1f7d4da21a added dual for complete lattice
haftmann
parents: 32642
diff changeset
   801
  bot ("\<bottom>")
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   802
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   803
lemmas mem_simps =
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   804
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   805
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   806
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
21669
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
   807
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   808
end