author | wenzelm |
Mon, 01 Aug 2016 22:36:47 +0200 | |
changeset 63576 | ba972a7dbeba |
parent 63575 | b9bd9e61fd63 |
child 63820 | 9f004fbf9d5c |
permissions | -rw-r--r-- |
63575 | 1 |
(* Title: HOL/Complete_Lattices.thy |
2 |
Author: Tobias Nipkow |
|
3 |
Author: Lawrence C Paulson |
|
4 |
Author: Markus Wenzel |
|
5 |
Author: Florian Haftmann |
|
6 |
*) |
|
11979 | 7 |
|
60758 | 8 |
section \<open>Complete lattices\<close> |
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
|
9 |
|
44860
56101fa00193
renamed theory Complete_Lattice to Complete_Lattices, in accordance with Lattices, Orderings etc.
haftmann
parents:
44845
diff
changeset
|
10 |
theory Complete_Lattices |
63575 | 11 |
imports Fun |
32139 | 12 |
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
|
13 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
14 |
notation |
34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset
|
15 |
less_eq (infix "\<sqsubseteq>" 50) and |
46691 | 16 |
less (infix "\<sqsubset>" 50) |
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
|
17 |
|
32139 | 18 |
|
60758 | 19 |
subsection \<open>Syntactic infimum and supremum operations\<close> |
32879 | 20 |
|
21 |
class Inf = |
|
63575 | 22 |
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) |
54257
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
23 |
begin |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
24 |
|
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
25 |
abbreviation INFIMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" |
63575 | 26 |
where "INFIMUM A f \<equiv> \<Sqinter>(f ` A)" |
56166 | 27 |
|
63575 | 28 |
lemma INF_image [simp]: "INFIMUM (f ` A) g = INFIMUM A (g \<circ> f)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
29 |
by (simp add: image_comp) |
54259
71c701dc5bf9
add SUP and INF for conditionally complete lattices
hoelzl
parents:
54257
diff
changeset
|
30 |
|
63575 | 31 |
lemma INF_identity_eq [simp]: "INFIMUM A (\<lambda>x. x) = \<Sqinter>A" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
32 |
by simp |
56166 | 33 |
|
63575 | 34 |
lemma INF_id_eq [simp]: "INFIMUM A id = \<Sqinter>A" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
35 |
by simp |
56166 | 36 |
|
63575 | 37 |
lemma INF_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> INFIMUM A C = INFIMUM B D" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
38 |
by (simp add: image_def) |
54259
71c701dc5bf9
add SUP and INF for conditionally complete lattices
hoelzl
parents:
54257
diff
changeset
|
39 |
|
56248
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
40 |
lemma strong_INF_cong [cong]: |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
41 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> INFIMUM A C = INFIMUM B D" |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
42 |
unfolding simp_implies_def by (fact INF_cong) |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
43 |
|
54257
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
44 |
end |
32879 | 45 |
|
46 |
class Sup = |
|
63575 | 47 |
fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) |
54257
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
48 |
begin |
32879 | 49 |
|
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
50 |
abbreviation SUPREMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" |
63575 | 51 |
where "SUPREMUM A f \<equiv> \<Squnion>(f ` A)" |
56166 | 52 |
|
63575 | 53 |
lemma SUP_image [simp]: "SUPREMUM (f ` A) g = SUPREMUM A (g \<circ> f)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
54 |
by (simp add: image_comp) |
54259
71c701dc5bf9
add SUP and INF for conditionally complete lattices
hoelzl
parents:
54257
diff
changeset
|
55 |
|
63575 | 56 |
lemma SUP_identity_eq [simp]: "SUPREMUM A (\<lambda>x. x) = \<Squnion>A" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
57 |
by simp |
56166 | 58 |
|
63575 | 59 |
lemma SUP_id_eq [simp]: "SUPREMUM A id = \<Squnion>A" |
56166 | 60 |
by (simp add: id_def) |
61 |
||
63575 | 62 |
lemma SUP_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> SUPREMUM A C = SUPREMUM B D" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
63 |
by (simp add: image_def) |
54259
71c701dc5bf9
add SUP and INF for conditionally complete lattices
hoelzl
parents:
54257
diff
changeset
|
64 |
|
56248
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
65 |
lemma strong_SUP_cong [cong]: |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
66 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> SUPREMUM A C = SUPREMUM B D" |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
67 |
unfolding simp_implies_def by (fact SUP_cong) |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
68 |
|
54257
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
69 |
end |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
70 |
|
60758 | 71 |
text \<open> |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
72 |
Note: must use names @{const INFIMUM} and @{const SUPREMUM} here instead of |
61799 | 73 |
\<open>INF\<close> and \<open>SUP\<close> to allow the following syntax coexist |
54257
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
74 |
with the plain constant names. |
60758 | 75 |
\<close> |
54257
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
76 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
77 |
syntax (ASCII) |
54257
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
78 |
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10) |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
79 |
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10) |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
80 |
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10) |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
81 |
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10) |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
82 |
|
62048
fefd79f6b232
retain ASCII syntax for output, when HOL/Library/Lattice_Syntax is not present (amending e96292f32c3c);
wenzelm
parents:
61955
diff
changeset
|
83 |
syntax (output) |
fefd79f6b232
retain ASCII syntax for output, when HOL/Library/Lattice_Syntax is not present (amending e96292f32c3c);
wenzelm
parents:
61955
diff
changeset
|
84 |
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10) |
fefd79f6b232
retain ASCII syntax for output, when HOL/Library/Lattice_Syntax is not present (amending e96292f32c3c);
wenzelm
parents:
61955
diff
changeset
|
85 |
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10) |
fefd79f6b232
retain ASCII syntax for output, when HOL/Library/Lattice_Syntax is not present (amending e96292f32c3c);
wenzelm
parents:
61955
diff
changeset
|
86 |
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10) |
fefd79f6b232
retain ASCII syntax for output, when HOL/Library/Lattice_Syntax is not present (amending e96292f32c3c);
wenzelm
parents:
61955
diff
changeset
|
87 |
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10) |
fefd79f6b232
retain ASCII syntax for output, when HOL/Library/Lattice_Syntax is not present (amending e96292f32c3c);
wenzelm
parents:
61955
diff
changeset
|
88 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
89 |
syntax |
54257
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
90 |
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
91 |
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
92 |
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
93 |
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) |
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
94 |
|
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
95 |
translations |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
96 |
"\<Sqinter>x y. B" \<rightleftharpoons> "\<Sqinter>x. \<Sqinter>y. B" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
97 |
"\<Sqinter>x. B" \<rightleftharpoons> "CONST INFIMUM CONST UNIV (\<lambda>x. B)" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
98 |
"\<Sqinter>x. B" \<rightleftharpoons> "\<Sqinter>x \<in> CONST UNIV. B" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
99 |
"\<Sqinter>x\<in>A. B" \<rightleftharpoons> "CONST INFIMUM A (\<lambda>x. B)" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
100 |
"\<Squnion>x y. B" \<rightleftharpoons> "\<Squnion>x. \<Squnion>y. B" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
101 |
"\<Squnion>x. B" \<rightleftharpoons> "CONST SUPREMUM CONST UNIV (\<lambda>x. B)" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
102 |
"\<Squnion>x. B" \<rightleftharpoons> "\<Squnion>x \<in> CONST UNIV. B" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
103 |
"\<Squnion>x\<in>A. B" \<rightleftharpoons> "CONST SUPREMUM A (\<lambda>x. B)" |
54257
5c7a3b6b05a9
generalize SUP and INF to the syntactic type classes Sup and Inf
hoelzl
parents:
54147
diff
changeset
|
104 |
|
60758 | 105 |
print_translation \<open> |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
106 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFIMUM} @{syntax_const "_INF"}, |
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
107 |
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPREMUM} @{syntax_const "_SUP"}] |
61799 | 108 |
\<close> \<comment> \<open>to avoid eta-contraction of body\<close> |
46691 | 109 |
|
62048
fefd79f6b232
retain ASCII syntax for output, when HOL/Library/Lattice_Syntax is not present (amending e96292f32c3c);
wenzelm
parents:
61955
diff
changeset
|
110 |
|
60758 | 111 |
subsection \<open>Abstract complete lattices\<close> |
32139 | 112 |
|
60758 | 113 |
text \<open>A complete lattice always has a bottom and a top, |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
114 |
so we include them into the following type class, |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
115 |
along with assumptions that define bottom and top |
60758 | 116 |
in terms of infimum and supremum.\<close> |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
117 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
118 |
class complete_lattice = lattice + Inf + Sup + bot + top + |
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
|
119 |
assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" |
63575 | 120 |
and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" |
121 |
and Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" |
|
122 |
and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" |
|
123 |
and Inf_empty [simp]: "\<Sqinter>{} = \<top>" |
|
124 |
and Sup_empty [simp]: "\<Squnion>{} = \<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
|
125 |
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
|
126 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
127 |
subclass bounded_lattice |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
128 |
proof |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
129 |
fix a |
63575 | 130 |
show "\<bottom> \<le> a" |
131 |
by (auto intro: Sup_least simp only: Sup_empty [symmetric]) |
|
132 |
show "a \<le> \<top>" |
|
133 |
by (auto intro: Inf_greatest simp only: Inf_empty [symmetric]) |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
134 |
qed |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
135 |
|
63575 | 136 |
lemma dual_complete_lattice: "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
137 |
by (auto intro!: class.complete_lattice.intro dual_lattice) |
63575 | 138 |
(unfold_locales, (fact Inf_empty Sup_empty Sup_upper Sup_least Inf_lower Inf_greatest)+) |
32678 | 139 |
|
44040 | 140 |
end |
141 |
||
142 |
context complete_lattice |
|
143 |
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
|
144 |
|
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
145 |
lemma Sup_eqI: |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
146 |
"(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> (\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> \<Squnion>A = x" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
147 |
by (blast intro: antisym Sup_least Sup_upper) |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
148 |
|
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
149 |
lemma Inf_eqI: |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
150 |
"(\<And>i. i \<in> A \<Longrightarrow> x \<le> i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x) \<Longrightarrow> \<Sqinter>A = x" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
151 |
by (blast intro: antisym Inf_greatest Inf_lower) |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
152 |
|
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
153 |
lemma SUP_eqI: |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
154 |
"(\<And>i. i \<in> A \<Longrightarrow> f i \<le> x) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> (\<Squnion>i\<in>A. f i) = x" |
56166 | 155 |
using Sup_eqI [of "f ` A" x] by auto |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
156 |
|
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
157 |
lemma INF_eqI: |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
158 |
"(\<And>i. i \<in> A \<Longrightarrow> x \<le> f i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<ge> y) \<Longrightarrow> x \<ge> y) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) = x" |
56166 | 159 |
using Inf_eqI [of "f ` A" x] by auto |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
160 |
|
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
161 |
lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i" |
56166 | 162 |
using Inf_lower [of _ "f ` A"] by simp |
44040 | 163 |
|
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
164 |
lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)" |
56166 | 165 |
using Inf_greatest [of "f ` A"] by auto |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
166 |
|
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
167 |
lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)" |
56166 | 168 |
using Sup_upper [of _ "f ` A"] by simp |
44040 | 169 |
|
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
170 |
lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u" |
56166 | 171 |
using Sup_least [of "f ` A"] by auto |
44040 | 172 |
|
173 |
lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v" |
|
174 |
using Inf_lower [of u A] by auto |
|
175 |
||
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
176 |
lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u" |
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
177 |
using INF_lower [of i A f] by auto |
44040 | 178 |
|
179 |
lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A" |
|
180 |
using Sup_upper [of u A] by auto |
|
181 |
||
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
182 |
lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)" |
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
183 |
using SUP_upper [of i A f] by auto |
44040 | 184 |
|
44918 | 185 |
lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)" |
44040 | 186 |
by (auto intro: Inf_greatest dest: Inf_lower) |
187 |
||
44918 | 188 |
lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)" |
56166 | 189 |
using le_Inf_iff [of _ "f ` A"] by simp |
44040 | 190 |
|
44918 | 191 |
lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)" |
44040 | 192 |
by (auto intro: Sup_least dest: Sup_upper) |
193 |
||
44918 | 194 |
lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)" |
56166 | 195 |
using Sup_le_iff [of "f ` A"] by simp |
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
|
196 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
197 |
lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
198 |
by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
199 |
|
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
200 |
lemma INF_insert [simp]: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFIMUM A f" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
201 |
by (simp cong del: strong_INF_cong) |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
202 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
203 |
lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
204 |
by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52141
diff
changeset
|
205 |
|
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
206 |
lemma SUP_insert [simp]: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPREMUM A f" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
207 |
by (simp cong del: strong_SUP_cong) |
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
|
208 |
|
44067 | 209 |
lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
210 |
by (simp cong del: strong_INF_cong) |
44040 | 211 |
|
44067 | 212 |
lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
213 |
by (simp cong del: strong_SUP_cong) |
44040 | 214 |
|
63575 | 215 |
lemma Inf_UNIV [simp]: "\<Sqinter>UNIV = \<bottom>" |
44040 | 216 |
by (auto intro!: antisym Inf_lower) |
41080 | 217 |
|
63575 | 218 |
lemma Sup_UNIV [simp]: "\<Squnion>UNIV = \<top>" |
44040 | 219 |
by (auto intro!: antisym Sup_upper) |
41080 | 220 |
|
44040 | 221 |
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}" |
222 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
|
223 |
||
224 |
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}" |
|
225 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
|
226 |
||
43899 | 227 |
lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B" |
228 |
by (auto intro: Inf_greatest Inf_lower) |
|
229 |
||
230 |
lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B" |
|
231 |
by (auto intro: Sup_least Sup_upper) |
|
232 |
||
38705 | 233 |
lemma Inf_mono: |
41971 | 234 |
assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b" |
43741 | 235 |
shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B" |
38705 | 236 |
proof (rule Inf_greatest) |
237 |
fix b assume "b \<in> B" |
|
41971 | 238 |
with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast |
60758 | 239 |
from \<open>a \<in> A\<close> have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower) |
240 |
with \<open>a \<sqsubseteq> b\<close> show "\<Sqinter>A \<sqsubseteq> b" by auto |
|
38705 | 241 |
qed |
242 |
||
63575 | 243 |
lemma INF_mono: "(\<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)" |
56166 | 244 |
using Inf_mono [of "g ` B" "f ` A"] by auto |
44041 | 245 |
|
41082 | 246 |
lemma Sup_mono: |
41971 | 247 |
assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b" |
43741 | 248 |
shows "\<Squnion>A \<sqsubseteq> \<Squnion>B" |
41082 | 249 |
proof (rule Sup_least) |
250 |
fix a assume "a \<in> A" |
|
41971 | 251 |
with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast |
60758 | 252 |
from \<open>b \<in> B\<close> have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper) |
253 |
with \<open>a \<sqsubseteq> b\<close> show "a \<sqsubseteq> \<Squnion>B" by auto |
|
41082 | 254 |
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
|
255 |
|
63575 | 256 |
lemma SUP_mono: "(\<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)" |
56166 | 257 |
using Sup_mono [of "f ` A" "g ` B"] by auto |
44041 | 258 |
|
63575 | 259 |
lemma INF_superset_mono: "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)" |
61799 | 260 |
\<comment> \<open>The last inclusion is POSITIVE!\<close> |
44041 | 261 |
by (blast intro: INF_mono dest: subsetD) |
262 |
||
63575 | 263 |
lemma SUP_subset_mono: "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)" |
44041 | 264 |
by (blast intro: SUP_mono dest: subsetD) |
265 |
||
43868 | 266 |
lemma Inf_less_eq: |
267 |
assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u" |
|
268 |
and "A \<noteq> {}" |
|
269 |
shows "\<Sqinter>A \<sqsubseteq> u" |
|
270 |
proof - |
|
60758 | 271 |
from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast |
272 |
moreover from \<open>v \<in> A\<close> assms(1) have "v \<sqsubseteq> u" by blast |
|
43868 | 273 |
ultimately show ?thesis by (rule Inf_lower2) |
274 |
qed |
|
275 |
||
276 |
lemma less_eq_Sup: |
|
277 |
assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v" |
|
278 |
and "A \<noteq> {}" |
|
279 |
shows "u \<sqsubseteq> \<Squnion>A" |
|
280 |
proof - |
|
60758 | 281 |
from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast |
282 |
moreover from \<open>v \<in> A\<close> assms(1) have "u \<sqsubseteq> v" by blast |
|
43868 | 283 |
ultimately show ?thesis by (rule Sup_upper2) |
284 |
qed |
|
285 |
||
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
286 |
lemma INF_eq: |
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
287 |
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j" |
63575 | 288 |
and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
289 |
shows "INFIMUM A f = INFIMUM B g" |
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
290 |
by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+ |
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
291 |
|
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56166
diff
changeset
|
292 |
lemma SUP_eq: |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
293 |
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j" |
63575 | 294 |
and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
295 |
shows "SUPREMUM A f = SUPREMUM B g" |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
296 |
by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+ |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
297 |
|
43899 | 298 |
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)" |
43868 | 299 |
by (auto intro: Inf_greatest Inf_lower) |
300 |
||
43899 | 301 |
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B " |
43868 | 302 |
by (auto intro: Sup_least Sup_upper) |
303 |
||
304 |
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B" |
|
305 |
by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2) |
|
306 |
||
63575 | 307 |
lemma INF_union: "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
308 |
by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower) |
44041 | 309 |
|
43868 | 310 |
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B" |
311 |
by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2) |
|
312 |
||
63575 | 313 |
lemma SUP_union: "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
314 |
by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper) |
44041 | 315 |
|
316 |
lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)" |
|
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
317 |
by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono) |
44041 | 318 |
|
63575 | 319 |
lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)" |
320 |
(is "?L = ?R") |
|
44918 | 321 |
proof (rule antisym) |
63575 | 322 |
show "?L \<le> ?R" |
323 |
by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono) |
|
324 |
show "?R \<le> ?L" |
|
325 |
by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper) |
|
44918 | 326 |
qed |
44041 | 327 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
328 |
lemma Inf_top_conv [simp]: |
43868 | 329 |
"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
330 |
"\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
331 |
proof - |
|
332 |
show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
333 |
proof |
|
334 |
assume "\<forall>x\<in>A. x = \<top>" |
|
335 |
then have "A = {} \<or> A = {\<top>}" by auto |
|
44919 | 336 |
then show "\<Sqinter>A = \<top>" by auto |
43868 | 337 |
next |
338 |
assume "\<Sqinter>A = \<top>" |
|
339 |
show "\<forall>x\<in>A. x = \<top>" |
|
340 |
proof (rule ccontr) |
|
341 |
assume "\<not> (\<forall>x\<in>A. x = \<top>)" |
|
342 |
then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast |
|
343 |
then obtain B where "A = insert x B" by blast |
|
60758 | 344 |
with \<open>\<Sqinter>A = \<top>\<close> \<open>x \<noteq> \<top>\<close> show False by simp |
43868 | 345 |
qed |
346 |
qed |
|
347 |
then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto |
|
348 |
qed |
|
349 |
||
44918 | 350 |
lemma INF_top_conv [simp]: |
56166 | 351 |
"(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
352 |
"\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
|
353 |
using Inf_top_conv [of "B ` A"] by simp_all |
|
44041 | 354 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
355 |
lemma Sup_bot_conv [simp]: |
63575 | 356 |
"\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" |
357 |
"\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" |
|
44920 | 358 |
using dual_complete_lattice |
359 |
by (rule complete_lattice.Inf_top_conv)+ |
|
43868 | 360 |
|
44918 | 361 |
lemma SUP_bot_conv [simp]: |
63575 | 362 |
"(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
363 |
"\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
|
56166 | 364 |
using Sup_bot_conv [of "B ` A"] by simp_all |
44041 | 365 |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
366 |
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
367 |
by (auto intro: antisym INF_lower 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
|
368 |
|
43870 | 369 |
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
370 |
by (auto intro: antisym SUP_upper SUP_least) |
43870 | 371 |
|
44918 | 372 |
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>" |
44921 | 373 |
by (cases "A = {}") simp_all |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
374 |
|
44918 | 375 |
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>" |
44921 | 376 |
by (cases "A = {}") simp_all |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
377 |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
378 |
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)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
379 |
by (iprover intro: INF_lower INF_greatest order_trans antisym) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
380 |
|
43870 | 381 |
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)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
382 |
by (iprover intro: SUP_upper SUP_least order_trans antisym) |
43870 | 383 |
|
43871 | 384 |
lemma INF_absorb: |
43868 | 385 |
assumes "k \<in> I" |
386 |
shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)" |
|
387 |
proof - |
|
388 |
from assms obtain J where "I = insert k J" by blast |
|
56166 | 389 |
then show ?thesis by simp |
43868 | 390 |
qed |
391 |
||
43871 | 392 |
lemma SUP_absorb: |
393 |
assumes "k \<in> I" |
|
394 |
shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)" |
|
395 |
proof - |
|
396 |
from assms obtain J where "I = insert k J" by blast |
|
56166 | 397 |
then show ?thesis by simp |
43871 | 398 |
qed |
399 |
||
63575 | 400 |
lemma INF_inf_const1: "I \<noteq> {} \<Longrightarrow> (INF i:I. inf x (f i)) = inf x (INF i:I. f i)" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
401 |
by (intro antisym INF_greatest inf_mono order_refl INF_lower) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
402 |
(auto intro: INF_lower2 le_infI2 intro!: INF_mono) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
403 |
|
63575 | 404 |
lemma INF_inf_const2: "I \<noteq> {} \<Longrightarrow> (INF i:I. inf (f i) x) = inf (INF i:I. f i) x" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
405 |
using INF_inf_const1[of I x f] by (simp add: inf_commute) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
406 |
|
63575 | 407 |
lemma INF_constant: "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)" |
44921 | 408 |
by simp |
43868 | 409 |
|
63575 | 410 |
lemma SUP_constant: "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)" |
44921 | 411 |
by simp |
43871 | 412 |
|
43943 | 413 |
lemma less_INF_D: |
63575 | 414 |
assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" |
415 |
shows "y < f i" |
|
43943 | 416 |
proof - |
60758 | 417 |
note \<open>y < (\<Sqinter>i\<in>A. f i)\<close> |
418 |
also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using \<open>i \<in> A\<close> |
|
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
419 |
by (rule INF_lower) |
43943 | 420 |
finally show "y < f i" . |
421 |
qed |
|
422 |
||
423 |
lemma SUP_lessD: |
|
63575 | 424 |
assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" |
425 |
shows "f i < y" |
|
43943 | 426 |
proof - |
63575 | 427 |
have "f i \<le> (\<Squnion>i\<in>A. f i)" |
428 |
using \<open>i \<in> A\<close> by (rule SUP_upper) |
|
60758 | 429 |
also note \<open>(\<Squnion>i\<in>A. f i) < y\<close> |
43943 | 430 |
finally show "f i < y" . |
431 |
qed |
|
432 |
||
63575 | 433 |
lemma INF_UNIV_bool_expand: "(\<Sqinter>b. A b) = A True \<sqinter> A False" |
56166 | 434 |
by (simp add: UNIV_bool inf_commute) |
43868 | 435 |
|
63575 | 436 |
lemma SUP_UNIV_bool_expand: "(\<Squnion>b. A b) = A True \<squnion> A False" |
56166 | 437 |
by (simp add: UNIV_bool sup_commute) |
43871 | 438 |
|
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
439 |
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
440 |
by (blast intro: Sup_upper2 Inf_lower ex_in_conv) |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
441 |
|
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
442 |
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFIMUM A f \<le> SUPREMUM A f" |
56166 | 443 |
using Inf_le_Sup [of "f ` A"] by simp |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
444 |
|
63575 | 445 |
lemma INF_eq_const: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFIMUM I f = x" |
54414
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
446 |
by (auto intro: INF_eqI) |
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
447 |
|
63575 | 448 |
lemma SUP_eq_const: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPREMUM I f = x" |
56248
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
449 |
by (auto intro: SUP_eqI) |
54414
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
450 |
|
63575 | 451 |
lemma INF_eq_iff: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<le> c) \<Longrightarrow> INFIMUM I f = c \<longleftrightarrow> (\<forall>i\<in>I. f i = c)" |
56248
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
452 |
using INF_eq_const [of I f c] INF_lower [of _ I f] |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
453 |
by (auto intro: antisym cong del: strong_INF_cong) |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
454 |
|
63575 | 455 |
lemma SUP_eq_iff: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> c \<le> f i) \<Longrightarrow> SUPREMUM I f = c \<longleftrightarrow> (\<forall>i\<in>I. f i = c)" |
56248
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
456 |
using SUP_eq_const [of I f c] SUP_upper [of _ I f] |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
457 |
by (auto intro: antisym cong del: strong_SUP_cong) |
54414
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
458 |
|
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
|
459 |
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
|
460 |
|
44024 | 461 |
class complete_distrib_lattice = complete_lattice + |
44039 | 462 |
assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" |
63575 | 463 |
and inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)" |
44024 | 464 |
begin |
465 |
||
63575 | 466 |
lemma sup_INF: "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)" |
63172 | 467 |
by (simp add: sup_Inf) |
44039 | 468 |
|
63575 | 469 |
lemma inf_SUP: "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)" |
63172 | 470 |
by (simp add: inf_Sup) |
44039 | 471 |
|
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
472 |
lemma dual_complete_distrib_lattice: |
44845 | 473 |
"class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" |
44024 | 474 |
apply (rule class.complete_distrib_lattice.intro) |
63575 | 475 |
apply (fact dual_complete_lattice) |
44024 | 476 |
apply (rule class.complete_distrib_lattice_axioms.intro) |
63575 | 477 |
apply (simp_all add: inf_Sup sup_Inf) |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
478 |
done |
44024 | 479 |
|
63575 | 480 |
subclass distrib_lattice |
481 |
proof |
|
44024 | 482 |
fix a b c |
63575 | 483 |
have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" by (rule sup_Inf) |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
484 |
then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by simp |
44024 | 485 |
qed |
486 |
||
63575 | 487 |
lemma Inf_sup: "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)" |
44039 | 488 |
by (simp add: sup_Inf sup_commute) |
489 |
||
63575 | 490 |
lemma Sup_inf: "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)" |
44039 | 491 |
by (simp add: inf_Sup inf_commute) |
492 |
||
63575 | 493 |
lemma INF_sup: "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)" |
44039 | 494 |
by (simp add: sup_INF sup_commute) |
495 |
||
63575 | 496 |
lemma SUP_inf: "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)" |
44039 | 497 |
by (simp add: inf_SUP inf_commute) |
498 |
||
63575 | 499 |
lemma Inf_sup_eq_top_iff: "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)" |
44039 | 500 |
by (simp only: Inf_sup INF_top_conv) |
501 |
||
63575 | 502 |
lemma Sup_inf_eq_bot_iff: "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)" |
44039 | 503 |
by (simp only: Sup_inf SUP_bot_conv) |
504 |
||
63575 | 505 |
lemma INF_sup_distrib2: "(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)" |
44039 | 506 |
by (subst INF_commute) (simp add: sup_INF INF_sup) |
507 |
||
63575 | 508 |
lemma SUP_inf_distrib2: "(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)" |
44039 | 509 |
by (subst SUP_commute) (simp add: inf_SUP SUP_inf) |
510 |
||
56074 | 511 |
context |
512 |
fixes f :: "'a \<Rightarrow> 'b::complete_lattice" |
|
513 |
assumes "mono f" |
|
514 |
begin |
|
515 |
||
63575 | 516 |
lemma mono_Inf: "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)" |
60758 | 517 |
using \<open>mono f\<close> by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD) |
56074 | 518 |
|
63575 | 519 |
lemma mono_Sup: "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)" |
60758 | 520 |
using \<open>mono f\<close> by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD) |
56074 | 521 |
|
63575 | 522 |
lemma mono_INF: "f (INF i : I. A i) \<le> (INF x : I. f (A x))" |
60758 | 523 |
by (intro complete_lattice_class.INF_greatest monoD[OF \<open>mono f\<close>] INF_lower) |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
58889
diff
changeset
|
524 |
|
63575 | 525 |
lemma mono_SUP: "(SUP x : I. f (A x)) \<le> f (SUP i : I. A i)" |
60758 | 526 |
by (intro complete_lattice_class.SUP_least monoD[OF \<open>mono f\<close>] SUP_upper) |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
58889
diff
changeset
|
527 |
|
56074 | 528 |
end |
529 |
||
44024 | 530 |
end |
531 |
||
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
532 |
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice |
43873 | 533 |
begin |
534 |
||
43943 | 535 |
lemma dual_complete_boolean_algebra: |
44845 | 536 |
"class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus" |
63575 | 537 |
by (rule class.complete_boolean_algebra.intro, |
538 |
rule dual_complete_distrib_lattice, |
|
539 |
rule dual_boolean_algebra) |
|
43943 | 540 |
|
63575 | 541 |
lemma uminus_Inf: "- (\<Sqinter>A) = \<Squnion>(uminus ` A)" |
43873 | 542 |
proof (rule antisym) |
543 |
show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)" |
|
544 |
by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp |
|
545 |
show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A" |
|
546 |
by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto |
|
547 |
qed |
|
548 |
||
44041 | 549 |
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
550 |
by (simp add: uminus_Inf image_image) |
44041 | 551 |
|
63575 | 552 |
lemma uminus_Sup: "- (\<Squnion>A) = \<Sqinter>(uminus ` A)" |
43873 | 553 |
proof - |
63575 | 554 |
have "\<Squnion>A = - \<Sqinter>(uminus ` A)" |
555 |
by (simp add: image_image uminus_INF) |
|
43873 | 556 |
then show ?thesis by simp |
557 |
qed |
|
63575 | 558 |
|
43873 | 559 |
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
560 |
by (simp add: uminus_Sup image_image) |
43873 | 561 |
|
562 |
end |
|
563 |
||
43940 | 564 |
class complete_linorder = linorder + complete_lattice |
565 |
begin |
|
566 |
||
43943 | 567 |
lemma dual_complete_linorder: |
44845 | 568 |
"class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" |
43943 | 569 |
by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder) |
570 |
||
51386 | 571 |
lemma complete_linorder_inf_min: "inf = min" |
51540
eea5c4ca4a0e
explicit sublocale dependency for Min/Max yields more appropriate Min/Max prefix for a couple of facts
haftmann
parents:
51489
diff
changeset
|
572 |
by (auto intro: antisym simp add: min_def fun_eq_iff) |
51386 | 573 |
|
574 |
lemma complete_linorder_sup_max: "sup = max" |
|
51540
eea5c4ca4a0e
explicit sublocale dependency for Min/Max yields more appropriate Min/Max prefix for a couple of facts
haftmann
parents:
51489
diff
changeset
|
575 |
by (auto intro: antisym simp add: max_def fun_eq_iff) |
51386 | 576 |
|
63575 | 577 |
lemma Inf_less_iff: "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)" |
63172 | 578 |
by (simp add: not_le [symmetric] le_Inf_iff) |
43940 | 579 |
|
63575 | 580 |
lemma INF_less_iff: "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)" |
63172 | 581 |
by (simp add: Inf_less_iff [of "f ` A"]) |
44041 | 582 |
|
63575 | 583 |
lemma less_Sup_iff: "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)" |
63172 | 584 |
by (simp add: not_le [symmetric] Sup_le_iff) |
43940 | 585 |
|
63575 | 586 |
lemma less_SUP_iff: "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)" |
63172 | 587 |
by (simp add: less_Sup_iff [of _ "f ` A"]) |
43940 | 588 |
|
63575 | 589 |
lemma Sup_eq_top_iff [simp]: "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)" |
43943 | 590 |
proof |
591 |
assume *: "\<Squnion>A = \<top>" |
|
63575 | 592 |
show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" |
593 |
unfolding * [symmetric] |
|
43943 | 594 |
proof (intro allI impI) |
63575 | 595 |
fix x |
596 |
assume "x < \<Squnion>A" |
|
597 |
then show "\<exists>i\<in>A. x < i" |
|
63172 | 598 |
by (simp add: less_Sup_iff) |
43943 | 599 |
qed |
600 |
next |
|
601 |
assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i" |
|
602 |
show "\<Squnion>A = \<top>" |
|
603 |
proof (rule ccontr) |
|
604 |
assume "\<Squnion>A \<noteq> \<top>" |
|
63575 | 605 |
with top_greatest [of "\<Squnion>A"] have "\<Squnion>A < \<top>" |
606 |
unfolding le_less by auto |
|
607 |
with * have "\<Squnion>A < \<Squnion>A" |
|
608 |
unfolding less_Sup_iff by auto |
|
43943 | 609 |
then show False by auto |
610 |
qed |
|
611 |
qed |
|
612 |
||
63575 | 613 |
lemma SUP_eq_top_iff [simp]: "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)" |
56166 | 614 |
using Sup_eq_top_iff [of "f ` A"] by simp |
44041 | 615 |
|
63575 | 616 |
lemma Inf_eq_bot_iff [simp]: "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)" |
44920 | 617 |
using dual_complete_linorder |
618 |
by (rule complete_linorder.Sup_eq_top_iff) |
|
43943 | 619 |
|
63575 | 620 |
lemma INF_eq_bot_iff [simp]: "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)" |
56166 | 621 |
using Inf_eq_bot_iff [of "f ` A"] by simp |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
622 |
|
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
623 |
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
624 |
proof safe |
63575 | 625 |
fix y |
626 |
assume "x \<ge> \<Sqinter>A" "y > x" |
|
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
627 |
then have "y > \<Sqinter>A" by auto |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
628 |
then show "\<exists>a\<in>A. y > a" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
629 |
unfolding Inf_less_iff . |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
630 |
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower) |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
631 |
|
63575 | 632 |
lemma INF_le_iff: "INFIMUM A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)" |
56166 | 633 |
using Inf_le_iff [of "f ` A"] by simp |
634 |
||
635 |
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)" |
|
636 |
proof safe |
|
63575 | 637 |
fix y |
638 |
assume "x \<le> \<Squnion>A" "y < x" |
|
56166 | 639 |
then have "y < \<Squnion>A" by auto |
640 |
then show "\<exists>a\<in>A. y < a" |
|
641 |
unfolding less_Sup_iff . |
|
642 |
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper) |
|
643 |
||
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
644 |
lemma le_SUP_iff: "x \<le> SUPREMUM A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)" |
56166 | 645 |
using le_Sup_iff [of _ "f ` A"] by simp |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
646 |
|
51386 | 647 |
subclass complete_distrib_lattice |
648 |
proof |
|
649 |
fix a and B |
|
650 |
show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)" |
|
651 |
by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper) |
|
652 |
(auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff |
|
653 |
le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min) |
|
654 |
qed |
|
655 |
||
43940 | 656 |
end |
657 |
||
51341
8c10293e7ea7
complete_linorder is also a complete_distrib_lattice
hoelzl
parents:
51328
diff
changeset
|
658 |
|
60758 | 659 |
subsection \<open>Complete lattice on @{typ bool}\<close> |
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
|
660 |
|
44024 | 661 |
instantiation bool :: 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
|
662 |
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
|
663 |
|
63575 | 664 |
definition [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> 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
|
665 |
|
63575 | 666 |
definition [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> 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
|
667 |
|
63575 | 668 |
instance |
669 |
by standard (auto intro: bool_induct) |
|
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
|
670 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
671 |
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
|
672 |
|
63575 | 673 |
lemma not_False_in_image_Ball [simp]: "False \<notin> P ` A \<longleftrightarrow> Ball A P" |
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
674 |
by auto |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
675 |
|
63575 | 676 |
lemma True_in_image_Bex [simp]: "True \<in> P ` A \<longleftrightarrow> Bex A P" |
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
677 |
by auto |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
678 |
|
63575 | 679 |
lemma INF_bool_eq [simp]: "INFIMUM = Ball" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
680 |
by (simp add: fun_eq_iff) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
681 |
|
63575 | 682 |
lemma SUP_bool_eq [simp]: "SUPREMUM = Bex" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
683 |
by (simp add: fun_eq_iff) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
684 |
|
63575 | 685 |
instance bool :: complete_boolean_algebra |
686 |
by standard (auto intro: bool_induct) |
|
44024 | 687 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
688 |
|
60758 | 689 |
subsection \<open>Complete lattice on @{typ "_ \<Rightarrow> _"}\<close> |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
690 |
|
57197
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
691 |
instantiation "fun" :: (type, Inf) Inf |
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
|
692 |
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
|
693 |
|
63575 | 694 |
definition "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)" |
41080 | 695 |
|
63575 | 696 |
lemma Inf_apply [simp, code]: "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)" |
41080 | 697 |
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
|
698 |
|
57197
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
699 |
instance .. |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
700 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
701 |
end |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
702 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
703 |
instantiation "fun" :: (type, Sup) Sup |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
704 |
begin |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
705 |
|
63575 | 706 |
definition "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)" |
41080 | 707 |
|
63575 | 708 |
lemma Sup_apply [simp, code]: "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)" |
41080 | 709 |
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
|
710 |
|
57197
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
711 |
instance .. |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
712 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
713 |
end |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
714 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
715 |
instantiation "fun" :: (type, complete_lattice) complete_lattice |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
716 |
begin |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
717 |
|
63575 | 718 |
instance |
719 |
by standard (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper 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
|
720 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
721 |
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
|
722 |
|
63575 | 723 |
lemma INF_apply [simp]: "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)" |
56166 | 724 |
using Inf_apply [of "f ` A"] by (simp add: comp_def) |
38705 | 725 |
|
63575 | 726 |
lemma SUP_apply [simp]: "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)" |
56166 | 727 |
using Sup_apply [of "f ` A"] by (simp add: comp_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
|
728 |
|
63575 | 729 |
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice |
730 |
by standard (auto simp add: inf_Sup sup_Inf fun_eq_iff image_image) |
|
44024 | 731 |
|
43873 | 732 |
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra .. |
733 |
||
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
734 |
|
60758 | 735 |
subsection \<open>Complete lattice on unary and binary predicates\<close> |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
736 |
|
63575 | 737 |
lemma Inf1_I: "(\<And>P. P \<in> A \<Longrightarrow> P a) \<Longrightarrow> (\<Sqinter>A) a" |
46884 | 738 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
739 |
|
63575 | 740 |
lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
741 |
by simp |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
742 |
|
63575 | 743 |
lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
744 |
by simp |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
745 |
|
63575 | 746 |
lemma Inf2_I: "(\<And>r. r \<in> A \<Longrightarrow> r a b) \<Longrightarrow> (\<Sqinter>A) a b" |
46884 | 747 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
748 |
|
63575 | 749 |
lemma Inf1_D: "(\<Sqinter>A) a \<Longrightarrow> P \<in> A \<Longrightarrow> P a" |
46884 | 750 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
751 |
|
63575 | 752 |
lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
753 |
by simp |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
754 |
|
63575 | 755 |
lemma Inf2_D: "(\<Sqinter>A) a b \<Longrightarrow> r \<in> A \<Longrightarrow> r a b" |
46884 | 756 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
757 |
|
63575 | 758 |
lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
759 |
by simp |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
760 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
761 |
lemma Inf1_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
762 |
assumes "(\<Sqinter>A) a" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
763 |
obtains "P a" | "P \<notin> A" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
764 |
using assms by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
765 |
|
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
766 |
lemma INF1_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
767 |
assumes "(\<Sqinter>x\<in>A. B x) b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
768 |
obtains "B a b" | "a \<notin> A" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
769 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
770 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
771 |
lemma Inf2_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
772 |
assumes "(\<Sqinter>A) a b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
773 |
obtains "r a b" | "r \<notin> A" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
774 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
775 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
776 |
lemma INF2_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
777 |
assumes "(\<Sqinter>x\<in>A. B x) b c" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
778 |
obtains "B a b c" | "a \<notin> A" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
779 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
780 |
|
63575 | 781 |
lemma Sup1_I: "P \<in> A \<Longrightarrow> P a \<Longrightarrow> (\<Squnion>A) a" |
46884 | 782 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
783 |
|
63575 | 784 |
lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
785 |
by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
786 |
|
63575 | 787 |
lemma Sup2_I: "r \<in> A \<Longrightarrow> r a b \<Longrightarrow> (\<Squnion>A) a b" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
788 |
by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
789 |
|
63575 | 790 |
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c" |
46884 | 791 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
792 |
|
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
793 |
lemma Sup1_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
794 |
assumes "(\<Squnion>A) a" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
795 |
obtains P where "P \<in> A" and "P a" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
796 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
797 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
798 |
lemma SUP1_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
799 |
assumes "(\<Squnion>x\<in>A. B x) b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
800 |
obtains x where "x \<in> A" and "B x b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
801 |
using assms by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
802 |
|
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
803 |
lemma Sup2_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
804 |
assumes "(\<Squnion>A) a b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
805 |
obtains r where "r \<in> A" "r a b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
806 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
807 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
808 |
lemma SUP2_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
809 |
assumes "(\<Squnion>x\<in>A. B x) b c" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
810 |
obtains x where "x \<in> A" "B x b c" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
811 |
using assms by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
812 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
813 |
|
60758 | 814 |
subsection \<open>Complete lattice on @{typ "_ set"}\<close> |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
815 |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
816 |
instantiation "set" :: (type) complete_lattice |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
817 |
begin |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
818 |
|
63575 | 819 |
definition "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
820 |
|
63575 | 821 |
definition "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
822 |
|
63575 | 823 |
instance |
824 |
by standard (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def) |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
825 |
|
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
826 |
end |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
827 |
|
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
828 |
instance "set" :: (type) complete_boolean_algebra |
63575 | 829 |
by standard (auto simp add: Inf_set_def Sup_set_def image_def) |
830 |
||
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
|
831 |
|
60758 | 832 |
subsubsection \<open>Inter\<close> |
41082 | 833 |
|
61952 | 834 |
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" ("\<Inter>_" [900] 900) |
835 |
where "\<Inter>S \<equiv> \<Sqinter>S" |
|
63575 | 836 |
|
837 |
lemma Inter_eq: "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" |
|
41082 | 838 |
proof (rule set_eqI) |
839 |
fix x |
|
840 |
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" |
|
841 |
by auto |
|
842 |
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}" |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
843 |
by (simp add: Inf_set_def image_def) |
41082 | 844 |
qed |
845 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
846 |
lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)" |
41082 | 847 |
by (unfold Inter_eq) blast |
848 |
||
43741 | 849 |
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C" |
41082 | 850 |
by (simp add: Inter_eq) |
851 |
||
60758 | 852 |
text \<open> |
63575 | 853 |
\<^medskip> A ``destruct'' rule -- every @{term X} in @{term C} |
43741 | 854 |
contains @{term A} as an element, but @{prop "A \<in> X"} can hold when |
61799 | 855 |
@{prop "X \<in> C"} does not! This rule is analogous to \<open>spec\<close>. |
60758 | 856 |
\<close> |
41082 | 857 |
|
43741 | 858 |
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X" |
41082 | 859 |
by auto |
860 |
||
43741 | 861 |
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R" |
61799 | 862 |
\<comment> \<open>``Classical'' elimination rule -- does not require proving |
60758 | 863 |
@{prop "X \<in> C"}.\<close> |
63575 | 864 |
unfolding Inter_eq by blast |
41082 | 865 |
|
43741 | 866 |
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 867 |
by (fact Inf_lower) |
868 |
||
63575 | 869 |
lemma Inter_subset: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 870 |
by (fact Inf_less_eq) |
41082 | 871 |
|
61952 | 872 |
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> \<Inter>A" |
43740 | 873 |
by (fact Inf_greatest) |
41082 | 874 |
|
44067 | 875 |
lemma Inter_empty: "\<Inter>{} = UNIV" |
876 |
by (fact Inf_empty) (* already simp *) |
|
41082 | 877 |
|
44067 | 878 |
lemma Inter_UNIV: "\<Inter>UNIV = {}" |
879 |
by (fact Inf_UNIV) (* already simp *) |
|
41082 | 880 |
|
44920 | 881 |
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B" |
882 |
by (fact Inf_insert) (* already simp *) |
|
41082 | 883 |
|
884 |
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" |
|
43899 | 885 |
by (fact less_eq_Inf_inter) |
41082 | 886 |
|
887 |
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" |
|
43756 | 888 |
by (fact Inf_union_distrib) |
889 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
890 |
lemma Inter_UNIV_conv [simp]: |
43741 | 891 |
"\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
892 |
"UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
|
43801 | 893 |
by (fact Inf_top_conv)+ |
41082 | 894 |
|
43741 | 895 |
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B" |
43899 | 896 |
by (fact Inf_superset_mono) |
41082 | 897 |
|
898 |
||
60758 | 899 |
subsubsection \<open>Intersections of families\<close> |
41082 | 900 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
901 |
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
902 |
where "INTER \<equiv> INFIMUM" |
41082 | 903 |
|
60758 | 904 |
text \<open> |
61799 | 905 |
Note: must use name @{const INTER} here instead of \<open>INT\<close> |
43872 | 906 |
to allow the following syntax coexist with the plain constant name. |
60758 | 907 |
\<close> |
43872 | 908 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
909 |
syntax (ASCII) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
910 |
"_INTER1" :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3INT _./ _)" [0, 10] 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
911 |
"_INTER" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10) |
41082 | 912 |
|
913 |
syntax (latex output) |
|
62789 | 914 |
"_INTER1" :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
915 |
"_INTER" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
916 |
|
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
917 |
syntax |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
918 |
"_INTER1" :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
919 |
"_INTER" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10) |
41082 | 920 |
|
921 |
translations |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
922 |
"\<Inter>x y. B" \<rightleftharpoons> "\<Inter>x. \<Inter>y. B" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
923 |
"\<Inter>x. B" \<rightleftharpoons> "CONST INTER CONST UNIV (\<lambda>x. B)" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
924 |
"\<Inter>x. B" \<rightleftharpoons> "\<Inter>x \<in> CONST UNIV. B" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
925 |
"\<Inter>x\<in>A. B" \<rightleftharpoons> "CONST INTER A (\<lambda>x. B)" |
41082 | 926 |
|
60758 | 927 |
print_translation \<open> |
42284 | 928 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}] |
61799 | 929 |
\<close> \<comment> \<open>to avoid eta-contraction of body\<close> |
41082 | 930 |
|
63575 | 931 |
lemma INTER_eq: "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" |
56166 | 932 |
by (auto intro!: INF_eqI) |
41082 | 933 |
|
43817 | 934 |
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)" |
56166 | 935 |
using Inter_iff [of _ "B ` A"] by simp |
41082 | 936 |
|
43817 | 937 |
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
938 |
by auto |
41082 | 939 |
|
43852 | 940 |
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a" |
41082 | 941 |
by auto |
942 |
||
43852 | 943 |
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" |
61799 | 944 |
\<comment> \<open>"Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}.\<close> |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
945 |
by auto |
41082 | 946 |
|
947 |
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" |
|
948 |
by blast |
|
949 |
||
950 |
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" |
|
951 |
by blast |
|
952 |
||
43817 | 953 |
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
954 |
by (fact INF_lower) |
41082 | 955 |
|
43817 | 956 |
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
957 |
by (fact INF_greatest) |
41082 | 958 |
|
44067 | 959 |
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
960 |
by (fact INF_empty) |
43854 | 961 |
|
43817 | 962 |
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" |
43872 | 963 |
by (fact INF_absorb) |
41082 | 964 |
|
43854 | 965 |
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)" |
41082 | 966 |
by (fact le_INF_iff) |
967 |
||
968 |
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
|
969 |
by (fact INF_insert) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
970 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
971 |
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
|
972 |
by (fact INF_union) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
973 |
|
63575 | 974 |
lemma INT_insert_distrib: "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
975 |
by blast |
43854 | 976 |
|
41082 | 977 |
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
|
978 |
by (fact INF_constant) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
979 |
|
44920 | 980 |
lemma INTER_UNIV_conv: |
63575 | 981 |
"(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)" |
982 |
"((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" |
|
44920 | 983 |
by (fact INF_top_conv)+ (* already simp *) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
984 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
985 |
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False" |
43873 | 986 |
by (fact INF_UNIV_bool_expand) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
987 |
|
63575 | 988 |
lemma INT_anti_mono: "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)" |
61799 | 989 |
\<comment> \<open>The last inclusion is POSITIVE!\<close> |
43940 | 990 |
by (fact INF_superset_mono) |
41082 | 991 |
|
992 |
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" |
|
993 |
by blast |
|
994 |
||
43817 | 995 |
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)" |
41082 | 996 |
by blast |
997 |
||
998 |
||
60758 | 999 |
subsubsection \<open>Union\<close> |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1000 |
|
61952 | 1001 |
abbreviation Union :: "'a set set \<Rightarrow> 'a set" ("\<Union>_" [900] 900) |
1002 |
where "\<Union>S \<equiv> \<Squnion>S" |
|
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1003 |
|
63575 | 1004 |
lemma Union_eq: "\<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
|
1005 |
proof (rule set_eqI) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1006 |
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
|
1007 |
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
|
1008 |
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
|
1009 |
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1010 |
by (simp add: Sup_set_def image_def) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1011 |
qed |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1012 |
|
63575 | 1013 |
lemma Union_iff [simp]: "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1014 |
by (unfold Union_eq) blast |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1015 |
|
63575 | 1016 |
lemma UnionI [intro]: "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C" |
61799 | 1017 |
\<comment> \<open>The order of the premises presupposes that @{term C} is rigid; |
60758 | 1018 |
@{term A} may be flexible.\<close> |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1019 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1020 |
|
63575 | 1021 |
lemma UnionE [elim!]: "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
|
1022 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1023 |
|
43817 | 1024 |
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A" |
43901 | 1025 |
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
|
1026 |
|
43817 | 1027 |
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C" |
43901 | 1028 |
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
|
1029 |
|
44920 | 1030 |
lemma Union_empty: "\<Union>{} = {}" |
1031 |
by (fact Sup_empty) (* already simp *) |
|
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
|
1032 |
|
44920 | 1033 |
lemma Union_UNIV: "\<Union>UNIV = UNIV" |
1034 |
by (fact Sup_UNIV) (* already simp *) |
|
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
|
1035 |
|
44920 | 1036 |
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B" |
1037 |
by (fact Sup_insert) (* already simp *) |
|
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
|
1038 |
|
43817 | 1039 |
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B" |
43901 | 1040 |
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
|
1041 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1042 |
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B" |
43901 | 1043 |
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
|
1044 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1045 |
lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
44920 | 1046 |
by (fact Sup_bot_conv) (* already simp *) |
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
|
1047 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1048 |
lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
44920 | 1049 |
by (fact Sup_bot_conv) (* already simp *) |
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
|
1050 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1051 |
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
|
1052 |
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
|
1053 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1054 |
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
|
1055 |
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
|
1056 |
|
43817 | 1057 |
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" |
43901 | 1058 |
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
|
1059 |
|
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63365
diff
changeset
|
1060 |
lemma Union_subsetI: "(\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> B \<and> x \<subseteq> y) \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63365
diff
changeset
|
1061 |
by blast |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1062 |
|
63575 | 1063 |
|
60758 | 1064 |
subsubsection \<open>Unions of families\<close> |
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
|
1065 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1066 |
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1067 |
where "UNION \<equiv> SUPREMUM" |
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
|
1068 |
|
60758 | 1069 |
text \<open> |
61799 | 1070 |
Note: must use name @{const UNION} here instead of \<open>UN\<close> |
43872 | 1071 |
to allow the following syntax coexist with the plain constant name. |
60758 | 1072 |
\<close> |
43872 | 1073 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1074 |
syntax (ASCII) |
35115 | 1075 |
"_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
|
1076 |
"_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
|
1077 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1078 |
syntax (latex output) |
62789 | 1079 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
1080 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(\<open>unbreakable\<close>\<^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
|
1081 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1082 |
syntax |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1083 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1084 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1085 |
|
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
|
1086 |
translations |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1087 |
"\<Union>x y. B" \<rightleftharpoons> "\<Union>x. \<Union>y. B" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1088 |
"\<Union>x. B" \<rightleftharpoons> "CONST UNION CONST UNIV (\<lambda>x. B)" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1089 |
"\<Union>x. B" \<rightleftharpoons> "\<Union>x \<in> CONST UNIV. B" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1090 |
"\<Union>x\<in>A. B" \<rightleftharpoons> "CONST UNION A (\<lambda>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
|
1091 |
|
60758 | 1092 |
text \<open> |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1093 |
Note the difference between ordinary syntax of indexed |
61799 | 1094 |
unions and intersections (e.g.\ \<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>) |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1095 |
and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}. |
60758 | 1096 |
\<close> |
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
|
1097 |
|
60758 | 1098 |
print_translation \<open> |
42284 | 1099 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}] |
61799 | 1100 |
\<close> \<comment> \<open>to avoid eta-contraction of body\<close> |
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
|
1101 |
|
63575 | 1102 |
lemma UNION_eq: "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}" |
56166 | 1103 |
by (auto intro!: SUP_eqI) |
44920 | 1104 |
|
63575 | 1105 |
lemma bind_UNION [code]: "Set.bind A f = UNION A f" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1106 |
by (simp add: bind_def UNION_eq) |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1107 |
|
63575 | 1108 |
lemma member_bind [simp]: "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f " |
46036 | 1109 |
by (simp add: bind_UNION) |
1110 |
||
60585 | 1111 |
lemma Union_SetCompr_eq: "\<Union>{f x| x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}" |
60307
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60172
diff
changeset
|
1112 |
by blast |
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60172
diff
changeset
|
1113 |
|
46036 | 1114 |
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)" |
56166 | 1115 |
using Union_iff [of _ "B ` A"] by simp |
11979 | 1116 |
|
43852 | 1117 |
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)" |
61799 | 1118 |
\<comment> \<open>The order of the premises presupposes that @{term A} is rigid; |
60758 | 1119 |
@{term b} may be flexible.\<close> |
11979 | 1120 |
by auto |
1121 |
||
43852 | 1122 |
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" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1123 |
by auto |
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
|
1124 |
|
43817 | 1125 |
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
1126 |
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
|
1127 |
|
43817 | 1128 |
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
1129 |
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
|
1130 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1131 |
lemma Collect_bex_eq: "{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
|
1132 |
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
|
1133 |
|
43817 | 1134 |
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
|
1135 |
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
|
1136 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1137 |
lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1138 |
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
|
1139 |
|
44920 | 1140 |
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}" |
1141 |
by (fact SUP_bot) (* already simp *) |
|
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
|
1142 |
|
43817 | 1143 |
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
|
1144 |
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
|
1145 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1146 |
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
|
1147 |
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
|
1148 |
|
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1149 |
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
|
1150 |
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
|
1151 |
|
43967 | 1152 |
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)" |
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
|
1153 |
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
|
1154 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1155 |
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" |
35629 | 1156 |
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
|
1157 |
|
f645b51e8e54
set 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 |
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
|
1159 |
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
|
1160 |
|
43944 | 1161 |
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. 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
|
1162 |
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
|
1163 |
|
44920 | 1164 |
lemma UNION_empty_conv: |
43817 | 1165 |
"{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
1166 |
"(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
|
44920 | 1167 |
by (fact SUP_bot_conv)+ (* already simp *) |
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
|
1168 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1169 |
lemma Collect_ex_eq: "{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
|
1170 |
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
|
1171 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1172 |
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
|
1173 |
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
|
1174 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1175 |
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
|
1176 |
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
|
1177 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1178 |
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)" |
62390 | 1179 |
by safe (auto simp add: if_split_mem2) |
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
|
1180 |
|
43817 | 1181 |
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
|
1182 |
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
|
1183 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1184 |
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
|
1185 |
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
|
1186 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1187 |
lemma UN_mono: |
43817 | 1188 |
"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
|
1189 |
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)" |
43940 | 1190 |
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
|
1191 |
|
43817 | 1192 |
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
|
1193 |
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
|
1194 |
|
43817 | 1195 |
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
|
1196 |
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
|
1197 |
|
43817 | 1198 |
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})" |
61799 | 1199 |
\<comment> \<open>NOT suitable for rewriting\<close> |
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
|
1200 |
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
|
1201 |
|
43817 | 1202 |
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)" |
1203 |
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
|
1204 |
|
45013 | 1205 |
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A" |
1206 |
by blast |
|
1207 |
||
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62789
diff
changeset
|
1208 |
lemma inj_on_image: "inj_on f (\<Union>A) \<Longrightarrow> inj_on (op ` f) A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62789
diff
changeset
|
1209 |
unfolding inj_on_def by blast |
11979 | 1210 |
|
63575 | 1211 |
|
60758 | 1212 |
subsubsection \<open>Distributive laws\<close> |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1213 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1214 |
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1215 |
by (fact inf_Sup) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1216 |
|
44039 | 1217 |
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)" |
1218 |
by (fact sup_Inf) |
|
1219 |
||
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1220 |
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)" |
44039 | 1221 |
by (fact Sup_inf) |
1222 |
||
1223 |
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)" |
|
1224 |
by (rule sym) (rule INF_inf_distrib) |
|
1225 |
||
1226 |
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)" |
|
1227 |
by (rule sym) (rule SUP_sup_distrib) |
|
1228 |
||
63575 | 1229 |
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" (* FIXME drop *) |
56166 | 1230 |
by (simp add: INT_Int_distrib) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1231 |
|
63575 | 1232 |
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" (* FIXME drop *) |
61799 | 1233 |
\<comment> \<open>Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:\<close> |
1234 |
\<comment> \<open>Union of a family of unions\<close> |
|
56166 | 1235 |
by (simp add: UN_Un_distrib) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1236 |
|
44039 | 1237 |
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)" |
1238 |
by (fact sup_INF) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1239 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1240 |
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)" |
61799 | 1241 |
\<comment> \<open>Halmos, Naive Set Theory, page 35.\<close> |
44039 | 1242 |
by (fact inf_SUP) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1243 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1244 |
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)" |
44039 | 1245 |
by (fact SUP_inf_distrib2) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1246 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1247 |
lemma 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)" |
44039 | 1248 |
by (fact INF_sup_distrib2) |
1249 |
||
1250 |
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" |
|
1251 |
by (fact Sup_inf_eq_bot_iff) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1252 |
|
61630 | 1253 |
lemma SUP_UNION: "(SUP x:(UN y:A. g y). f x) = (SUP y:A. SUP x:g y. f x :: _ :: complete_lattice)" |
63575 | 1254 |
by (rule order_antisym) (blast intro: SUP_least SUP_upper2)+ |
1255 |
||
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1256 |
|
60758 | 1257 |
subsection \<open>Injections and bijections\<close> |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1258 |
|
63575 | 1259 |
lemma inj_on_Inter: "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)" |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1260 |
unfolding inj_on_def by blast |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1261 |
|
63575 | 1262 |
lemma inj_on_INTER: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)) \<Longrightarrow> inj_on f (\<Inter>i \<in> I. A i)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1263 |
unfolding inj_on_def by safe simp |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1264 |
|
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1265 |
lemma inj_on_UNION_chain: |
63575 | 1266 |
assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" |
1267 |
and inj: "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)" |
|
60585 | 1268 |
shows "inj_on f (\<Union>i \<in> I. A i)" |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1269 |
proof - |
63575 | 1270 |
have "x = y" |
1271 |
if *: "i \<in> I" "j \<in> I" |
|
1272 |
and **: "x \<in> A i" "y \<in> A j" |
|
1273 |
and ***: "f x = f y" |
|
1274 |
for i j x y |
|
1275 |
using chain [OF *] |
|
1276 |
proof |
|
1277 |
assume "A i \<le> A j" |
|
1278 |
with ** have "x \<in> A j" by auto |
|
1279 |
with inj * ** *** show ?thesis |
|
1280 |
by (auto simp add: inj_on_def) |
|
1281 |
next |
|
1282 |
assume "A j \<le> A i" |
|
1283 |
with ** have "y \<in> A i" by auto |
|
1284 |
with inj * ** *** show ?thesis |
|
1285 |
by (auto simp add: inj_on_def) |
|
1286 |
qed |
|
1287 |
then show ?thesis |
|
1288 |
by (unfold inj_on_def UNION_eq) auto |
|
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1289 |
qed |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1290 |
|
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1291 |
lemma bij_betw_UNION_chain: |
63575 | 1292 |
assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" |
1293 |
and bij: "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)" |
|
60585 | 1294 |
shows "bij_betw f (\<Union>i \<in> I. A i) (\<Union>i \<in> I. A' i)" |
63575 | 1295 |
unfolding bij_betw_def |
63576 | 1296 |
proof safe |
63575 | 1297 |
have "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)" |
1298 |
using bij bij_betw_def[of f] by auto |
|
63576 | 1299 |
then show "inj_on f (UNION I A)" |
63575 | 1300 |
using chain inj_on_UNION_chain[of I A f] by auto |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1301 |
next |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1302 |
fix i x |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1303 |
assume *: "i \<in> I" "x \<in> A i" |
63576 | 1304 |
with bij have "f x \<in> A' i" |
1305 |
by (auto simp: bij_betw_def) |
|
1306 |
with * show "f x \<in> UNION I A'" by blast |
|
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1307 |
next |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1308 |
fix i x' |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1309 |
assume *: "i \<in> I" "x' \<in> A' i" |
63576 | 1310 |
with bij have "\<exists>x \<in> A i. x' = f x" |
1311 |
unfolding bij_betw_def by blast |
|
63575 | 1312 |
with * have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x" |
1313 |
by blast |
|
63576 | 1314 |
then show "x' \<in> f ` UNION I A" |
63575 | 1315 |
by blast |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1316 |
qed |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1317 |
|
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1318 |
(*injectivity's required. Left-to-right inclusion holds even if A is empty*) |
63575 | 1319 |
lemma image_INT: "inj_on f C \<Longrightarrow> \<forall>x\<in>A. B x \<subseteq> C \<Longrightarrow> j \<in> A \<Longrightarrow> f ` (INTER A B) = (INT x:A. f ` B x)" |
1320 |
by (auto simp add: inj_on_def) blast |
|
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1321 |
|
63575 | 1322 |
lemma bij_image_INT: "bij f \<Longrightarrow> f ` (INTER A B) = (INT x:A. f ` B x)" |
1323 |
apply (simp only: bij_def) |
|
1324 |
apply (simp only: inj_on_def surj_def) |
|
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1325 |
apply auto |
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1326 |
apply blast |
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1327 |
done |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1328 |
|
63575 | 1329 |
lemma UNION_fun_upd: "UNION J (A(i := B)) = UNION (J - {i}) A \<union> (if i \<in> J then B else {})" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1330 |
by (auto simp add: set_eq_iff) |
63365 | 1331 |
|
1332 |
lemma bij_betw_Pow: |
|
1333 |
assumes "bij_betw f A B" |
|
1334 |
shows "bij_betw (image f) (Pow A) (Pow B)" |
|
1335 |
proof - |
|
1336 |
from assms have "inj_on f A" |
|
1337 |
by (rule bij_betw_imp_inj_on) |
|
1338 |
then have "inj_on f (\<Union>Pow A)" |
|
1339 |
by simp |
|
1340 |
then have "inj_on (image f) (Pow A)" |
|
1341 |
by (rule inj_on_image) |
|
1342 |
then have "bij_betw (image f) (Pow A) (image f ` Pow A)" |
|
1343 |
by (rule inj_on_imp_bij_betw) |
|
1344 |
moreover from assms have "f ` A = B" |
|
1345 |
by (rule bij_betw_imp_surj_on) |
|
1346 |
then have "image f ` Pow A = Pow B" |
|
1347 |
by (rule image_Pow_surj) |
|
1348 |
ultimately show ?thesis by simp |
|
1349 |
qed |
|
1350 |
||
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1351 |
|
60758 | 1352 |
subsubsection \<open>Complement\<close> |
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
|
1353 |
|
43873 | 1354 |
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)" |
1355 |
by (fact uminus_INF) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1356 |
|
43873 | 1357 |
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)" |
1358 |
by (fact uminus_SUP) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1359 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1360 |
|
60758 | 1361 |
subsubsection \<open>Miniscoping and maxiscoping\<close> |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1362 |
|
63575 | 1363 |
text \<open>\<^medskip> Miniscoping: pushing in quantifiers and big Unions and Intersections.\<close> |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1364 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1365 |
lemma UN_simps [simp]: |
43817 | 1366 |
"\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1367 |
"\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))" |
43852 | 1368 |
"\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1369 |
"\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)" |
43852 | 1370 |
"\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))" |
1371 |
"\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)" |
|
1372 |
"\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))" |
|
1373 |
"\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)" |
|
1374 |
"\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)" |
|
43831 | 1375 |
"\<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
|
1376 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1377 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1378 |
lemma INT_simps [simp]: |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1379 |
"\<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)" |
43831 | 1380 |
"\<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 | 1381 |
"\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)" |
1382 |
"\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))" |
|
43817 | 1383 |
"\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)" |
43852 | 1384 |
"\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)" |
1385 |
"\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))" |
|
1386 |
"\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)" |
|
1387 |
"\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)" |
|
1388 |
"\<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
|
1389 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1390 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1391 |
lemma UN_ball_bex_simps [simp]: |
43852 | 1392 |
"\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)" |
43967 | 1393 |
"\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)" |
43852 | 1394 |
"\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)" |
1395 |
"\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1396 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1397 |
|
43943 | 1398 |
|
63575 | 1399 |
text \<open>\<^medskip> Maxiscoping: pulling out big Unions and Intersections.\<close> |
13860 | 1400 |
|
1401 |
lemma UN_extend_simps: |
|
43817 | 1402 |
"\<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)))" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1403 |
"\<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))" |
43852 | 1404 |
"\<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))" |
1405 |
"\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)" |
|
1406 |
"\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)" |
|
43817 | 1407 |
"\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)" |
1408 |
"\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)" |
|
43852 | 1409 |
"\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)" |
1410 |
"\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)" |
|
43831 | 1411 |
"\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)" |
13860 | 1412 |
by auto |
1413 |
||
1414 |
lemma INT_extend_simps: |
|
43852 | 1415 |
"\<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))" |
1416 |
"\<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))" |
|
1417 |
"\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))" |
|
1418 |
"\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))" |
|
43817 | 1419 |
"\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))" |
43852 | 1420 |
"\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)" |
1421 |
"\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)" |
|
1422 |
"\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)" |
|
1423 |
"\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)" |
|
1424 |
"\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)" |
|
13860 | 1425 |
by auto |
1426 |
||
60758 | 1427 |
text \<open>Finally\<close> |
43872 | 1428 |
|
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
|
1429 |
no_notation |
46691 | 1430 |
less_eq (infix "\<sqsubseteq>" 50) and |
1431 |
less (infix "\<sqsubset>" 50) |
|
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
|
1432 |
|
30596 | 1433 |
lemmas mem_simps = |
1434 |
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff |
|
1435 |
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff |
|
61799 | 1436 |
\<comment> \<open>Each of these has ALREADY been added \<open>[simp]\<close> above.\<close> |
21669 | 1437 |
|
11979 | 1438 |
end |