src/HOL/Complete_Lattice.thy
author haftmann
Thu, 21 Jul 2011 22:47:13 +0200
changeset 43943 e6928fc2332a
parent 43940 26ca0bad226a
child 43944 b1b436f75070
permissions -rw-r--r--
moved some lemmas
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:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
    36
  "class.complete_lattice Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>"
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 36364
diff changeset
    37
  by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)
34007
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
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    47
lemma Inf_empty [simp]:
34007
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
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    51
lemma Sup_empty [simp]:
34007
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
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    55
lemma Inf_UNIV [simp]:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    56
  "\<Sqinter>UNIV = \<bottom>"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    57
  by (simp add: Sup_Inf Sup_empty [symmetric])
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    58
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    59
lemma Sup_UNIV [simp]:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    60
  "\<Squnion>UNIV = \<top>"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    61
  by (simp add: Inf_Sup Inf_empty [symmetric])
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
    62
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
    63
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
    64
  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
    65
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
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
    67
  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
    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_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
    70
  "\<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
    71
  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
    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_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
    74
  "\<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
    75
  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
    76
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    77
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
    78
  "\<Sqinter>{a, b} = a \<sqinter> b"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
    79
  by (simp add: 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
    80
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
    81
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
    82
  "\<Squnion>{a, b} = a \<squnion> b"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
    83
  by (simp add: 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
    84
43754
09d455c93bf8 tuned notation
haftmann
parents: 43753
diff changeset
    85
lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
35629
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
    86
  by (auto intro: Inf_greatest dest: Inf_lower)
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
    87
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
    88
lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
    89
  by (auto intro: Sup_least dest: Sup_upper)
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
    90
43899
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
    91
lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
    92
  by (auto intro: Inf_greatest Inf_lower)
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
    93
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
    94
lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
    95
  by (auto intro: Sup_least Sup_upper)
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
    96
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
    97
lemma Inf_mono:
41971
a54e8e95fe96 add lemmas for SUP and INF
hoelzl
parents: 41082
diff changeset
    98
  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
    99
  shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   100
proof (rule Inf_greatest)
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   101
  fix b assume "b \<in> B"
41971
a54e8e95fe96 add lemmas for SUP and INF
hoelzl
parents: 41082
diff changeset
   102
  with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   103
  from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   104
  with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   105
qed
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   106
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   107
lemma Sup_mono:
41971
a54e8e95fe96 add lemmas for SUP and INF
hoelzl
parents: 41082
diff changeset
   108
  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   109
  shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   110
proof (rule Sup_least)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   111
  fix a assume "a \<in> A"
41971
a54e8e95fe96 add lemmas for SUP and INF
hoelzl
parents: 41082
diff changeset
   112
  with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   113
  from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   114
  with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   115
qed
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
   116
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   117
lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   118
  using Sup_upper [of u A] by auto
41971
a54e8e95fe96 add lemmas for SUP and INF
hoelzl
parents: 41082
diff changeset
   119
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   120
lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   121
  using Inf_lower [of u A] by auto
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   122
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   123
lemma Inf_less_eq:
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   124
  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   125
    and "A \<noteq> {}"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   126
  shows "\<Sqinter>A \<sqsubseteq> u"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   127
proof -
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   128
  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   129
  moreover with assms have "v \<sqsubseteq> u" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   130
  ultimately show ?thesis by (rule Inf_lower2)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   131
qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   132
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   133
lemma less_eq_Sup:
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   134
  assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   135
    and "A \<noteq> {}"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   136
  shows "u \<sqsubseteq> \<Squnion>A"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   137
proof -
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   138
  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   139
  moreover with assms have "u \<sqsubseteq> v" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   140
  ultimately show ?thesis by (rule Sup_upper2)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   141
qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   142
43899
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
   143
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   144
  by (auto intro: Inf_greatest Inf_lower)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   145
43899
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
   146
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   147
  by (auto intro: Sup_least Sup_upper)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   148
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   149
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   150
  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   151
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   152
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   153
  by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   154
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   155
lemma Inf_top_conv [no_atp]:
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   156
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   157
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   158
proof -
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   159
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   160
  proof
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   161
    assume "\<forall>x\<in>A. x = \<top>"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   162
    then have "A = {} \<or> A = {\<top>}" by auto
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   163
    then show "\<Sqinter>A = \<top>" by auto
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   164
  next
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   165
    assume "\<Sqinter>A = \<top>"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   166
    show "\<forall>x\<in>A. x = \<top>"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   167
    proof (rule ccontr)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   168
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   169
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   170
      then obtain B where "A = insert x B" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   171
      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   172
    qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   173
  qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   174
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   175
qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   176
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   177
lemma Sup_bot_conv [no_atp]:
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   178
  "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   179
  "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   180
proof -
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   181
  interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   182
    by (fact dual_complete_lattice)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   183
  from dual.Inf_top_conv show ?P and ?Q by simp_all
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   184
qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   185
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
   186
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   187
  INF_def: "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
   188
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   189
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   190
  SUP_def: "SUPR A f = \<Squnion> (f ` A)"
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   191
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   192
text {*
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   193
  Note: must use names @{const INFI} and @{const SUPR} here instead of
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   194
  @{text INF} and @{text SUP} to allow the following syntax coexist
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   195
  with the plain constant names.
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   196
*}
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   197
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
   198
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
   199
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
syntax
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   201
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   202
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   203
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   204
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   205
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   206
syntax (xsymbols)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   207
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   208
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   209
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   210
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 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
   211
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
translations
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   213
  "INF x y. B"   == "INF x. INF y. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   214
  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   215
  "INF x. B"     == "INF x:CONST UNIV. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   216
  "INF x:A. B"   == "CONST INFI A (%x. 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
   217
  "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
   218
  "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
   219
  "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
   220
  "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
   221
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   222
print_translation {*
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   223
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   224
    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   225
*} -- {* to avoid eta-contraction of body *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   226
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
   227
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
   228
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
   229
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   230
lemma INF_empty: "(\<Sqinter>x\<in>{}. f x) = \<top>"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   231
  by (simp add: INF_def)
41971
a54e8e95fe96 add lemmas for SUP and INF
hoelzl
parents: 41082
diff changeset
   232
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   233
lemma SUP_empty: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   234
  by (simp add: SUP_def)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   235
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   236
lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   237
  by (simp add: INF_def 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
   238
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   239
lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   240
  by (simp add: SUP_def Sup_insert)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   241
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   242
lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   243
  by (auto simp add: INF_def intro: Inf_lower)
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
   244
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   245
lemma le_SUP_I: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   246
  by (auto simp add: SUP_def intro: Sup_upper)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   247
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   248
lemma INF_leI2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   249
  using INF_leI [of i A f] by auto
41971
a54e8e95fe96 add lemmas for SUP and INF
hoelzl
parents: 41082
diff changeset
   250
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   251
lemma le_SUP_I2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   252
  using le_SUP_I [of i A f] by auto
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   253
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   254
lemma le_INF_I: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   255
  by (auto simp add: INF_def intro: 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
   256
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   257
lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   258
  by (auto simp add: SUP_def intro: Sup_least)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   259
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   260
lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> f i)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   261
  by (auto simp add: INF_def le_Inf_iff)
35629
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   262
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   263
lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. f i \<sqsubseteq> u)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   264
  by (auto simp add: SUP_def Sup_le_iff)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   265
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   266
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   267
  by (auto intro: antisym INF_leI le_INF_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
   268
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   269
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   270
  by (auto intro: antisym SUP_leI le_SUP_I)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   271
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   272
lemma INF_top: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   273
  by (cases "A = {}") (simp_all add: INF_empty)
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   274
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   275
lemma SUP_bot: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   276
  by (cases "A = {}") (simp_all add: SUP_empty)
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   277
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   278
lemma INF_cong:
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   279
  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   280
  by (simp add: INF_def image_def)
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   281
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   282
lemma SUP_cong:
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   283
  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   284
  by (simp add: SUP_def image_def)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   285
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   286
lemma INF_mono:
43753
fe5e846c0839 tuned notation
haftmann
parents: 43741
diff changeset
   287
  "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   288
  by (force intro!: Inf_mono simp: INF_def)
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   289
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   290
lemma SUP_mono:
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   291
  "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   292
  by (force intro!: Sup_mono simp: SUP_def)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   293
43899
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
   294
lemma INF_superset_mono:
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   295
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   296
  -- {* The last inclusion is POSITIVE! *}
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   297
  by (blast intro: INF_mono dest: subsetD)
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   298
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   299
lemma SUP_subset_mono:
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   300
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   301
  by (blast intro: SUP_mono dest: subsetD)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   302
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   303
lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   304
  by (iprover intro: INF_leI le_INF_I order_trans antisym)
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   305
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   306
lemma SUP_commute: "(\<Squnion>i\<in>A. \<Squnion>j\<in>B. f i j) = (\<Squnion>j\<in>B. \<Squnion>i\<in>A. f i j)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   307
  by (iprover intro: SUP_leI le_SUP_I order_trans antisym)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   308
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   309
lemma INF_absorb:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   310
  assumes "k \<in> I"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   311
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   312
proof -
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   313
  from assms obtain J where "I = insert k J" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   314
  then show ?thesis by (simp add: INF_insert)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   315
qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   316
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   317
lemma SUP_absorb:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   318
  assumes "k \<in> I"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   319
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   320
proof -
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   321
  from assms obtain J where "I = insert k J" by blast
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   322
  then show ?thesis by (simp add: SUP_insert)
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   323
qed
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   324
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   325
lemma INF_union:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   326
  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   327
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 le_INF_I INF_leI)
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   328
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   329
lemma SUP_union:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   330
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   331
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_leI le_SUP_I)
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   332
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   333
lemma INF_constant:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   334
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   335
  by (simp add: INF_empty)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   336
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   337
lemma SUP_constant:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   338
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   339
  by (simp add: SUP_empty)
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   340
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   341
lemma INF_eq:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   342
  "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   343
  by (simp add: INF_def image_def)
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   344
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   345
lemma SUP_eq:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   346
  "(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   347
  by (simp add: SUP_def image_def)
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   348
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   349
lemma INF_top_conv:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   350
 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   351
 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   352
  by (auto simp add: INF_def Inf_top_conv)
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   353
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   354
lemma SUP_bot_conv:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   355
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   356
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   357
  by (auto simp add: SUP_def Sup_bot_conv)
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   358
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   359
lemma less_INF_D:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   360
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   361
proof -
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   362
  note `y < (\<Sqinter>i\<in>A. f i)`
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   363
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   364
    by (rule INF_leI)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   365
  finally show "y < f i" .
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   366
qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   367
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   368
lemma SUP_lessD:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   369
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   370
proof -
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   371
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   372
    by (rule le_SUP_I)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   373
  also note `(\<Squnion>i\<in>A. f i) < y`
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   374
  finally show "f i < y" .
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   375
qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   376
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   377
lemma INF_UNIV_range:
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   378
  "(\<Sqinter>x. f x) = \<Sqinter>range f"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   379
  by (fact INF_def)
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   380
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   381
lemma SUP_UNIV_range:
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   382
  "(\<Squnion>x. f x) = \<Squnion>range f"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   383
  by (fact SUP_def)
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   384
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   385
lemma INF_UNIV_bool_expand:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   386
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   387
  by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   388
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   389
lemma SUP_UNIV_bool_expand:
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   390
  "(\<Squnion>b. A b) = A True \<squnion> A False"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   391
  by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   392
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
   393
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
   394
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   395
class complete_boolean_algebra = boolean_algebra + complete_lattice
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   396
begin
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   397
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   398
lemma dual_complete_boolean_algebra:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   399
  "class.complete_boolean_algebra Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   400
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_lattice, rule dual_boolean_algebra)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   401
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   402
lemma uminus_Inf:
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   403
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   404
proof (rule antisym)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   405
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   406
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   407
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   408
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   409
qed
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   410
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   411
lemma uminus_Sup:
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   412
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   413
proof -
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   414
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   415
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   416
qed
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   417
  
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   418
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   419
  by (simp add: INF_def SUP_def uminus_Inf image_image)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   420
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   421
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   422
  by (simp add: INF_def SUP_def uminus_Sup image_image)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   423
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   424
end
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   425
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   426
class complete_linorder = linorder + complete_lattice
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   427
begin
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   428
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   429
lemma dual_complete_linorder:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   430
  "class.complete_linorder Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   431
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   432
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   433
lemma Inf_less_iff:
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   434
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   435
  unfolding not_le [symmetric] le_Inf_iff by auto
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   436
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   437
lemma less_Sup_iff:
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   438
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   439
  unfolding not_le [symmetric] Sup_le_iff by auto
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   440
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   441
lemma INF_less_iff:
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   442
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   443
  unfolding INF_def Inf_less_iff by auto
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   444
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   445
lemma less_SUP_iff:
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   446
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   447
  unfolding SUP_def less_Sup_iff by auto
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   448
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   449
lemma Sup_eq_top_iff:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   450
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   451
proof
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   452
  assume *: "\<Squnion>A = \<top>"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   453
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   454
  proof (intro allI impI)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   455
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   456
      unfolding less_Sup_iff by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   457
  qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   458
next
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   459
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   460
  show "\<Squnion>A = \<top>"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   461
  proof (rule ccontr)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   462
    assume "\<Squnion>A \<noteq> \<top>"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   463
    with top_greatest [of "\<Squnion>A"]
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   464
    have "\<Squnion>A < \<top>" unfolding le_less by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   465
    then have "\<Squnion>A < \<Squnion>A"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   466
      using * unfolding less_Sup_iff by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   467
    then show False by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   468
  qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   469
qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   470
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   471
lemma Inf_eq_bot_iff:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   472
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   473
proof -
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   474
  interpret dual: complete_linorder Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   475
    by (fact dual_complete_linorder)
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   476
  from dual.Sup_eq_top_iff show ?thesis .
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   477
qed
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
   478
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   479
end
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   480
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   481
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   482
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
   483
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   484
instantiation bool :: complete_boolean_algebra
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
   485
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
   486
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   487
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   488
  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
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
   489
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   490
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   491
  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
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
   492
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   493
instance proof
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   494
qed (auto simp add: Inf_bool_def Sup_bool_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
   495
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   496
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
   497
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   498
lemma INF_bool_eq [simp]:
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   499
  "INFI = Ball"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   500
proof (rule ext)+
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   501
  fix A :: "'a set"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   502
  fix P :: "'a \<Rightarrow> bool"
43753
fe5e846c0839 tuned notation
haftmann
parents: 43741
diff changeset
   503
  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   504
    by (auto simp add: Ball_def INF_def Inf_bool_def)
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   505
qed
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   506
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   507
lemma SUP_bool_eq [simp]:
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   508
  "SUPR = Bex"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   509
proof (rule ext)+
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   510
  fix A :: "'a set"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   511
  fix P :: "'a \<Rightarrow> bool"
43753
fe5e846c0839 tuned notation
haftmann
parents: 43741
diff changeset
   512
  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   513
    by (auto simp add: Bex_def SUP_def Sup_bool_def)
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   514
qed
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   515
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
   516
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
   517
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
   518
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   519
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   520
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   521
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   522
lemma Inf_apply:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   523
  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   524
  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
   525
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   526
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   527
  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   528
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   529
lemma Sup_apply:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   530
  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   531
  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
   532
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   533
instance proof
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   534
qed (auto simp add: le_fun_def Inf_apply Sup_apply
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
   535
  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
   536
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   537
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
   538
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   539
lemma INF_apply:
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   540
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   541
  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   542
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   543
lemma SUP_apply:
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   544
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   545
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
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
   546
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   547
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   548
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
   549
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   550
subsection {* Inter *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   551
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   552
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   553
  "Inter S \<equiv> \<Sqinter>S"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   554
  
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   555
notation (xsymbols)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   556
  Inter  ("\<Inter>_" [90] 90)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   557
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   558
lemma Inter_eq:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   559
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   560
proof (rule set_eqI)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   561
  fix x
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   562
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   563
    by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   564
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   565
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   566
qed
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   567
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   568
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   569
  by (unfold Inter_eq) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   570
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   571
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   572
  by (simp add: Inter_eq)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   573
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   574
text {*
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   575
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   576
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   577
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   578
*}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   579
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   580
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   581
  by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   582
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   583
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   584
  -- {* ``Classical'' elimination rule -- does not require proving
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   585
    @{prop "X \<in> C"}. *}
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   586
  by (unfold Inter_eq) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   587
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   588
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
43740
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   589
  by (fact Inf_lower)
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   590
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   591
lemma Inter_subset:
43755
5e4a595e63fc tuned notation
haftmann
parents: 43754
diff changeset
   592
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
43740
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   593
  by (fact Inf_less_eq)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   594
43755
5e4a595e63fc tuned notation
haftmann
parents: 43754
diff changeset
   595
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
43740
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   596
  by (fact Inf_greatest)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   597
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   598
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
43739
4529a3c56609 more succinct proofs
haftmann
parents: 42284
diff changeset
   599
  by (fact Inf_binary [symmetric])
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   600
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   601
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   602
  by (fact Inf_empty)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   603
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   604
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
43739
4529a3c56609 more succinct proofs
haftmann
parents: 42284
diff changeset
   605
  by (fact Inf_UNIV)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   606
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   607
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
43739
4529a3c56609 more succinct proofs
haftmann
parents: 42284
diff changeset
   608
  by (fact Inf_insert)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   609
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   610
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
43899
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
   611
  by (fact less_eq_Inf_inter)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   612
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   613
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
43756
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   614
  by (fact Inf_union_distrib)
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   615
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   616
lemma Inter_UNIV_conv [simp, no_atp]:
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   617
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   618
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
43801
097732301fc0 more generalization towards complete lattices
haftmann
parents: 43756
diff changeset
   619
  by (fact Inf_top_conv)+
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   620
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   621
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
43899
60ef6abb2f92 avoid misunderstandable names
haftmann
parents: 43898
diff changeset
   622
  by (fact Inf_superset_mono)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   623
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   624
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   625
subsection {* Intersections of families *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   626
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   627
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   628
  "INTER \<equiv> INFI"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   629
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   630
text {*
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   631
  Note: must use name @{const INTER} here instead of @{text INT}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   632
  to allow the following syntax coexist with the plain constant name.
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   633
*}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   634
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   635
syntax
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   636
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   637
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   638
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   639
syntax (xsymbols)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   640
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   641
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   642
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   643
syntax (latex output)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   644
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   645
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   646
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   647
translations
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   648
  "INT x y. B"  == "INT x. INT y. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   649
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   650
  "INT x. B"    == "INT x:CONST UNIV. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   651
  "INT x:A. B"  == "CONST INTER A (%x. B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   652
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   653
print_translation {*
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   654
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   655
*} -- {* to avoid eta-contraction of body *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   656
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   657
lemma INTER_eq_Inter_image:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   658
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   659
  by (fact INF_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   660
  
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   661
lemma Inter_def:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   662
  "\<Inter>S = (\<Inter>x\<in>S. x)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   663
  by (simp add: INTER_eq_Inter_image image_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   664
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   665
lemma INTER_def:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   666
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   667
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   668
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   669
lemma Inter_image_eq [simp]:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   670
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   671
  by (rule sym) (fact INF_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   672
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   673
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   674
  by (unfold INTER_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   675
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   676
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   677
  by (unfold INTER_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   678
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   679
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   680
  by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   681
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   682
lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   683
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   684
  by (unfold INTER_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   685
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   686
lemma INT_cong [cong]:
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   687
  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   688
  by (fact INF_cong)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   689
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   690
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   691
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   692
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   693
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   694
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   695
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   696
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   697
  by (fact INF_leI)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   698
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   699
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   700
  by (fact le_INF_I)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   701
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   702
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   703
  by (fact INF_empty)
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   704
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   705
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   706
  by (fact INF_absorb)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   707
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   708
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   709
  by (fact le_INF_iff)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   710
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   711
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   712
  by (fact INF_insert)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   713
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   714
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   715
  by (fact INF_union)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   716
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   717
lemma INT_insert_distrib:
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   718
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   719
  by blast
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   720
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   721
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   722
  by (fact INF_constant)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   723
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   724
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   725
  -- {* Look: it has an \emph{existential} quantifier *}
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   726
  by (fact INF_eq)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   727
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   728
lemma INTER_UNIV_conv [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   729
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   730
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   731
  by (fact INF_top_conv)+
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   732
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   733
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   734
  by (fact INF_UNIV_bool_expand)
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   735
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   736
lemma INT_anti_mono:
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   737
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   738
  -- {* The last inclusion is POSITIVE! *}
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   739
  by (fact INF_superset_mono)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   740
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   741
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   742
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   743
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   744
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   745
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   746
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   747
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   748
subsection {* Union *}
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   749
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   750
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
   751
  "Union S \<equiv> \<Squnion>S"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   752
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   753
notation (xsymbols)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   754
  Union  ("\<Union>_" [90] 90)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   755
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
   756
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
   757
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 38705
diff changeset
   758
proof (rule set_eqI)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   759
  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
   760
  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
   761
    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
   762
  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
   763
    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
   764
qed
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   765
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   766
lemma Union_iff [simp, no_atp]:
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   767
  "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
   768
  by (unfold Union_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   769
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   770
lemma UnionI [intro]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   771
  "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
   772
  -- {* 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
   773
    @{term A} may be flexible. *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   774
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   775
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   776
lemma UnionE [elim!]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   777
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   778
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   779
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   780
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   781
  by (fact Sup_upper)
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
   782
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   783
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   784
  by (fact Sup_least)
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
   785
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
   786
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
   787
  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
   788
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   789
lemma Union_empty [simp]: "\<Union>{} = {}"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   790
  by (fact Sup_empty)
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
   791
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   792
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   793
  by (fact Sup_UNIV)
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
   794
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   795
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   796
  by (fact Sup_insert)
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
   797
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   798
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   799
  by (fact Sup_union_distrib)
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
   800
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
   801
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   802
  by (fact Sup_inter_less_eq)
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
   803
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   804
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   805
  by (fact Sup_bot_conv)
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
   806
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   807
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   808
  by (fact Sup_bot_conv)
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
   809
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   810
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" -- "FIXME generalize"
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
   811
  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
   812
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
   813
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
   814
  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
   815
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
   816
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
   817
  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
   818
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   819
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
43901
3ab6c30d256d proof tuning
haftmann
parents: 43900
diff changeset
   820
  by (fact Sup_subset_mono)
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
   821
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   822
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   823
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
   824
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   825
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
   826
  "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
   827
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   828
text {*
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   829
  Note: must use name @{const UNION} here instead of @{text UN}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   830
  to allow the following syntax coexist with the plain constant name.
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   831
*}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   832
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
   833
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   834
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   835
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 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
   836
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   837
syntax (xsymbols)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   838
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   839
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 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
   840
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   841
syntax (latex output)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   842
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   843
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 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
   844
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   845
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
   846
  "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
   847
  "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
   848
  "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
   849
  "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
   850
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   851
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
   852
  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
   853
  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
   854
  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
   855
  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
   856
  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
   857
  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
   858
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   859
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   860
print_translation {*
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   861
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   862
*} -- {* 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
   863
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
   864
lemma UNION_eq_Union_image:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   865
  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   866
  by (fact SUP_def)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   867
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   868
lemma Union_def:
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   869
  "\<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
   870
  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
   871
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   872
lemma UNION_def [no_atp]:
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
   873
  "(\<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
   874
  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
   875
  
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   876
lemma Union_image_eq [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   877
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   878
  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
   879
  
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   880
lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   881
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   882
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   883
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   884
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   885
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   886
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   887
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   888
lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   889
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   890
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   891
lemma UN_cong [cong]:
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   892
  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   893
  by (fact SUP_cong)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   894
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   895
lemma strong_UN_cong:
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   896
  "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   897
  by (unfold simp_implies_def) (fact UN_cong)
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   898
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   899
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
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
   900
  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
   901
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   902
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   903
  by (fact le_SUP_I)
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
   904
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   905
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   906
  by (fact SUP_leI)
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
   907
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   908
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
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
   909
  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
   910
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   911
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B 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
   912
  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
   913
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   914
lemma UN_empty [simp, no_atp]: "(\<Union>x\<in>{}. B x) = {}"
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   915
  by (fact SUP_empty)
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
   916
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
   917
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   918
  by (fact SUP_bot)
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
   919
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
   920
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
   921
  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
   922
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   923
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   924
  by (fact SUP_absorb)
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
   925
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
   926
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   927
  by (fact SUP_insert)
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
   928
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
   929
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)"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   930
  by (fact SUP_union)
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
   931
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   932
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)" -- "FIXME generalize"
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
   933
  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
   934
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
   935
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
   936
  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
   937
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
   938
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   939
  by (fact SUP_constant)
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
   940
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
   941
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   942
  by (fact SUP_eq)
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   943
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   944
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)" -- "FIXME generalize"
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
   945
  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
   946
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
   947
lemma UNION_empty_conv[simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   948
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   949
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   950
  by (fact SUP_bot_conv)+
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
   951
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   952
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
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
   953
  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
   954
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   955
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
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
   956
  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
   957
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   958
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
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
   959
  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
   960
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
   961
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
   962
  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
   963
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   964
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
43900
7162691e740b generalization; various notation and proof tuning
haftmann
parents: 43899
diff changeset
   965
  by (fact SUP_UNIV_bool_expand)
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
   966
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
   967
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
   968
  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
   969
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
   970
lemma UN_mono:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   971
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
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
   972
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
   973
  by (fact SUP_subset_mono)
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
   974
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   975
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` 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
   976
  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
   977
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   978
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B 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
   979
  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
   980
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   981
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
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
   982
  -- {* 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
   983
  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
   984
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   985
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   986
  by blast
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
   987
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   988
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   989
subsection {* Distributive laws *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   990
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   991
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
   992
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   993
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   994
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
   995
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   996
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   997
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   998
  -- {* 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
   999
  -- {* Union of a family of unions *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1000
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1001
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1002
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
  1003
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1004
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1005
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1006
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
  1007
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1008
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1009
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1010
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1011
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1012
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
  1013
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1014
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1015
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1016
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
  1017
  -- {* Halmos, Naive Set Theory, page 35. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1018
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1019
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1020
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
  1021
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1022
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1023
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
  1024
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1025
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1026
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
  1027
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1028
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1029
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
  1030
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
  1031
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1032
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1033
  by (fact uminus_INF)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1034
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1035
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1036
  by (fact uminus_SUP)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1037
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1038
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
  1039
subsection {* Miniscoping and maxiscoping *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1040
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1041
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1042
           and Intersections. *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1043
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1044
lemma UN_simps [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1045
  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1046
  "\<And>A B C. (\<Union>x\<in>C. A x \<union>  B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1047
  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1048
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter>B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1049
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1050
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1051
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1052
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1053
  "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1054
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1055
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1056
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1057
lemma INT_simps [simp]:
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1058
  "\<And>A B C. (\<Inter>x\<in>C. A x \<inter> B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) \<inter>B)"
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1059
  "\<And>A B C. (\<Inter>x\<in>C. A \<inter> B x) = (if C={} then UNIV else A \<inter>(\<Inter>x\<in>C. B x))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1060
  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1061
  "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1062
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1063
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1064
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1065
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1066
  "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1067
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1068
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1069
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
  1070
lemma ball_simps [simp,no_atp]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1071
  "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1072
  "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1073
  "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1074
  "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1075
  "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1076
  "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1077
  "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1078
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1079
  "\<And>A B P. (\<forall>x\<in> UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1080
  "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1081
  "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1082
  "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1083
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1084
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
  1085
lemma bex_simps [simp,no_atp]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1086
  "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1087
  "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1088
  "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1089
  "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1090
  "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1091
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1092
  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1093
  "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1094
  "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1095
  "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1096
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1097
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1098
lemma (in complete_linorder) INF_eq_bot_iff:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1099
  fixes f :: "'b \<Rightarrow> 'a"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1100
  shows "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1101
  unfolding INF_def Inf_eq_bot_iff by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1102
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1103
lemma (in complete_linorder) SUP_eq_top_iff:
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1104
  fixes f :: "'b \<Rightarrow> 'a"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1105
  shows "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1106
  unfolding SUP_def Sup_eq_top_iff by auto
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1107
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1108
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1109
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1110
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1111
lemma UN_extend_simps:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1112
  "\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1113
  "\<And>A B C. (\<Union>x\<in>C. A x) \<union>  B  = (if C={} then B else (\<Union>x\<in>C. A x \<union>  B))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1114
  "\<And>A B C. A \<union> (\<Union>x\<in>C. B x) = (if C={} then A else (\<Union>x\<in>C. A \<union> B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1115
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1116
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1117
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1118
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1119
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1120
  "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1121
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1122
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1123
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1124
lemma INT_extend_simps:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1125
  "\<And>A B C. (\<Inter>x\<in>C. A x) \<inter> B = (if C={} then B else (\<Inter>x\<in>C. A x \<inter> B))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1126
  "\<And>A B C. A \<inter> (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A \<inter> B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1127
  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1128
  "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1129
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1130
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1131
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1132
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1133
  "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1134
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1135
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1136
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1137
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1138
text {* Legacy names *}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1139
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1140
lemmas (in complete_lattice) INFI_def = INF_def
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1141
lemmas (in complete_lattice) SUPR_def = SUP_def
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1142
lemmas (in complete_lattice) le_SUPI = le_SUP_I
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1143
lemmas (in complete_lattice) le_SUPI2 = le_SUP_I2
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1144
lemmas (in complete_lattice) le_INFI = le_INF_I
43943
e6928fc2332a moved some lemmas
haftmann
parents: 43940
diff changeset
  1145
lemmas (in complete_lattice) less_INFD = less_INF_D
43940
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
  1146
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
  1147
lemma (in complete_lattice) INF_subset:
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
  1148
  "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
  1149
  by (rule INF_superset_mono) auto
26ca0bad226a class complete_linorder
haftmann
parents: 43901
diff changeset
  1150
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1151
lemmas INFI_apply = INF_apply
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1152
lemmas SUPR_apply = SUP_apply
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1153
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1154
text {* Finally *}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1155
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
  1156
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
  1157
  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
  1158
  less (infix "\<sqsubset>" 50) and
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1159
  bot ("\<bottom>") and
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1160
  top ("\<top>") and
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
  1161
  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
  1162
  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
  1163
  Inf  ("\<Sqinter>_" [900] 900) and
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1164
  Sup  ("\<Squnion>_" [900] 900)
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
  1165
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
  1166
no_syntax (xsymbols)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1167
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1168
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
  1169
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
  1170
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
  1171
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1172
lemmas mem_simps =
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1173
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1174
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1175
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
21669
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
  1176
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1177
end