author | haftmann |
Fri, 16 Jan 2009 14:58:11 +0100 | |
changeset 29509 | 1ff0f3f08a7b |
parent 29223 | e09c53289830 |
child 29580 | 117b88da143c |
permissions | -rw-r--r-- |
21249 | 1 |
(* Title: HOL/Lattices.thy |
2 |
ID: $Id$ |
|
3 |
Author: Tobias Nipkow |
|
4 |
*) |
|
5 |
||
22454 | 6 |
header {* Abstract lattices *} |
21249 | 7 |
|
8 |
theory Lattices |
|
26794 | 9 |
imports Fun |
21249 | 10 |
begin |
11 |
||
28562 | 12 |
subsection {* Lattices *} |
21249 | 13 |
|
25206 | 14 |
notation |
25382 | 15 |
less_eq (infix "\<sqsubseteq>" 50) and |
16 |
less (infix "\<sqsubset>" 50) |
|
25206 | 17 |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
18 |
class lower_semilattice = order + |
21249 | 19 |
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70) |
22737 | 20 |
assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x" |
21 |
and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y" |
|
21733 | 22 |
and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z" |
21249 | 23 |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
24 |
class upper_semilattice = order + |
21249 | 25 |
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65) |
22737 | 26 |
assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y" |
27 |
and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y" |
|
21733 | 28 |
and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x" |
26014 | 29 |
begin |
30 |
||
31 |
text {* Dual lattice *} |
|
32 |
||
33 |
lemma dual_lattice: |
|
34 |
"lower_semilattice (op \<ge>) (op >) sup" |
|
27682 | 35 |
by (rule lower_semilattice.intro, rule dual_order) |
36 |
(unfold_locales, simp_all add: sup_least) |
|
26014 | 37 |
|
38 |
end |
|
21249 | 39 |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
40 |
class lattice = lower_semilattice + upper_semilattice |
21249 | 41 |
|
25382 | 42 |
|
28562 | 43 |
subsubsection {* Intro and elim rules*} |
21733 | 44 |
|
45 |
context lower_semilattice |
|
46 |
begin |
|
21249 | 47 |
|
25062 | 48 |
lemma le_infI1[intro]: |
49 |
assumes "a \<sqsubseteq> x" |
|
50 |
shows "a \<sqinter> b \<sqsubseteq> x" |
|
51 |
proof (rule order_trans) |
|
25482 | 52 |
from assms show "a \<sqsubseteq> x" . |
53 |
show "a \<sqinter> b \<sqsubseteq> a" by simp |
|
25062 | 54 |
qed |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
55 |
lemmas (in -) [rule del] = le_infI1 |
21249 | 56 |
|
25062 | 57 |
lemma le_infI2[intro]: |
58 |
assumes "b \<sqsubseteq> x" |
|
59 |
shows "a \<sqinter> b \<sqsubseteq> x" |
|
60 |
proof (rule order_trans) |
|
25482 | 61 |
from assms show "b \<sqsubseteq> x" . |
62 |
show "a \<sqinter> b \<sqsubseteq> b" by simp |
|
25062 | 63 |
qed |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
64 |
lemmas (in -) [rule del] = le_infI2 |
21733 | 65 |
|
21734 | 66 |
lemma le_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b" |
21733 | 67 |
by(blast intro: inf_greatest) |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
68 |
lemmas (in -) [rule del] = le_infI |
21249 | 69 |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
70 |
lemma le_infE [elim!]: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P" |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
71 |
by (blast intro: order_trans) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
72 |
lemmas (in -) [rule del] = le_infE |
21249 | 73 |
|
21734 | 74 |
lemma le_inf_iff [simp]: |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
75 |
"x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)" |
21733 | 76 |
by blast |
77 |
||
21734 | 78 |
lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
79 |
by (blast intro: antisym dest: eq_iff [THEN iffD1]) |
21249 | 80 |
|
25206 | 81 |
lemma mono_inf: |
82 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>lower_semilattice" |
|
83 |
shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<le> f A \<sqinter> f B" |
|
84 |
by (auto simp add: mono_def intro: Lattices.inf_greatest) |
|
21733 | 85 |
|
25206 | 86 |
end |
21733 | 87 |
|
88 |
context upper_semilattice |
|
89 |
begin |
|
21249 | 90 |
|
21734 | 91 |
lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b" |
25062 | 92 |
by (rule order_trans) auto |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
93 |
lemmas (in -) [rule del] = le_supI1 |
21249 | 94 |
|
21734 | 95 |
lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b" |
25062 | 96 |
by (rule order_trans) auto |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
97 |
lemmas (in -) [rule del] = le_supI2 |
21733 | 98 |
|
21734 | 99 |
lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x" |
26014 | 100 |
by (blast intro: sup_least) |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
101 |
lemmas (in -) [rule del] = le_supI |
21249 | 102 |
|
21734 | 103 |
lemma le_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
104 |
by (blast intro: order_trans) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
105 |
lemmas (in -) [rule del] = le_supE |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
106 |
|
21734 | 107 |
lemma ge_sup_conv[simp]: |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
108 |
"x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)" |
21733 | 109 |
by blast |
110 |
||
21734 | 111 |
lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
112 |
by (blast intro: antisym dest: eq_iff [THEN iffD1]) |
21734 | 113 |
|
25206 | 114 |
lemma mono_sup: |
115 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>upper_semilattice" |
|
116 |
shows "mono f \<Longrightarrow> f A \<squnion> f B \<le> f (A \<squnion> B)" |
|
117 |
by (auto simp add: mono_def intro: Lattices.sup_least) |
|
21733 | 118 |
|
25206 | 119 |
end |
23878 | 120 |
|
21733 | 121 |
|
122 |
subsubsection{* Equational laws *} |
|
21249 | 123 |
|
21733 | 124 |
context lower_semilattice |
125 |
begin |
|
126 |
||
127 |
lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
128 |
by (blast intro: antisym) |
21733 | 129 |
|
130 |
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
131 |
by (blast intro: antisym) |
21733 | 132 |
|
133 |
lemma inf_idem[simp]: "x \<sqinter> x = x" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
134 |
by (blast intro: antisym) |
21733 | 135 |
|
136 |
lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
137 |
by (blast intro: antisym) |
21733 | 138 |
|
139 |
lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
140 |
by (blast intro: antisym) |
21733 | 141 |
|
142 |
lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
143 |
by (blast intro: antisym) |
21733 | 144 |
|
145 |
lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
146 |
by (blast intro: antisym) |
21733 | 147 |
|
148 |
lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem |
|
149 |
||
150 |
end |
|
151 |
||
152 |
||
153 |
context upper_semilattice |
|
154 |
begin |
|
21249 | 155 |
|
21733 | 156 |
lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
157 |
by (blast intro: antisym) |
21733 | 158 |
|
159 |
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
160 |
by (blast intro: antisym) |
21733 | 161 |
|
162 |
lemma sup_idem[simp]: "x \<squnion> x = x" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
163 |
by (blast intro: antisym) |
21733 | 164 |
|
165 |
lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
166 |
by (blast intro: antisym) |
21733 | 167 |
|
168 |
lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
169 |
by (blast intro: antisym) |
21733 | 170 |
|
171 |
lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
172 |
by (blast intro: antisym) |
21249 | 173 |
|
21733 | 174 |
lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
175 |
by (blast intro: antisym) |
21733 | 176 |
|
177 |
lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem |
|
178 |
||
179 |
end |
|
21249 | 180 |
|
21733 | 181 |
context lattice |
182 |
begin |
|
183 |
||
184 |
lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
185 |
by (blast intro: antisym inf_le1 inf_greatest sup_ge1) |
21733 | 186 |
|
187 |
lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
188 |
by (blast intro: antisym sup_ge1 sup_least inf_le1) |
21733 | 189 |
|
21734 | 190 |
lemmas ACI = inf_ACI sup_ACI |
191 |
||
22454 | 192 |
lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2 |
193 |
||
21734 | 194 |
text{* Towards distributivity *} |
21249 | 195 |
|
21734 | 196 |
lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
197 |
by blast |
21734 | 198 |
|
199 |
lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
200 |
by blast |
21734 | 201 |
|
202 |
||
203 |
text{* If you have one of them, you have them all. *} |
|
21249 | 204 |
|
21733 | 205 |
lemma distrib_imp1: |
21249 | 206 |
assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" |
207 |
shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" |
|
208 |
proof- |
|
209 |
have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb) |
|
210 |
also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc) |
|
211 |
also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)" |
|
212 |
by(simp add:inf_sup_absorb inf_commute) |
|
213 |
also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D) |
|
214 |
finally show ?thesis . |
|
215 |
qed |
|
216 |
||
21733 | 217 |
lemma distrib_imp2: |
21249 | 218 |
assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" |
219 |
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" |
|
220 |
proof- |
|
221 |
have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb) |
|
222 |
also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc) |
|
223 |
also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)" |
|
224 |
by(simp add:sup_inf_absorb sup_commute) |
|
225 |
also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D) |
|
226 |
finally show ?thesis . |
|
227 |
qed |
|
228 |
||
21734 | 229 |
(* seems unused *) |
230 |
lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z" |
|
231 |
by blast |
|
232 |
||
21733 | 233 |
end |
21249 | 234 |
|
235 |
||
24164 | 236 |
subsection {* Distributive lattices *} |
21249 | 237 |
|
22454 | 238 |
class distrib_lattice = lattice + |
21249 | 239 |
assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" |
240 |
||
21733 | 241 |
context distrib_lattice |
242 |
begin |
|
243 |
||
244 |
lemma sup_inf_distrib2: |
|
21249 | 245 |
"(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)" |
246 |
by(simp add:ACI sup_inf_distrib1) |
|
247 |
||
21733 | 248 |
lemma inf_sup_distrib1: |
21249 | 249 |
"x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" |
250 |
by(rule distrib_imp2[OF sup_inf_distrib1]) |
|
251 |
||
21733 | 252 |
lemma inf_sup_distrib2: |
21249 | 253 |
"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)" |
254 |
by(simp add:ACI inf_sup_distrib1) |
|
255 |
||
21733 | 256 |
lemmas distrib = |
21249 | 257 |
sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2 |
258 |
||
21733 | 259 |
end |
260 |
||
21249 | 261 |
|
22454 | 262 |
subsection {* Uniqueness of inf and sup *} |
263 |
||
22737 | 264 |
lemma (in lower_semilattice) inf_unique: |
22454 | 265 |
fixes f (infixl "\<triangle>" 70) |
25062 | 266 |
assumes le1: "\<And>x y. x \<triangle> y \<le> x" and le2: "\<And>x y. x \<triangle> y \<le> y" |
267 |
and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" |
|
22737 | 268 |
shows "x \<sqinter> y = x \<triangle> y" |
22454 | 269 |
proof (rule antisym) |
25062 | 270 |
show "x \<triangle> y \<le> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2) |
22454 | 271 |
next |
25062 | 272 |
have leI: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" by (blast intro: greatest) |
273 |
show "x \<sqinter> y \<le> x \<triangle> y" by (rule leI) simp_all |
|
22454 | 274 |
qed |
275 |
||
22737 | 276 |
lemma (in upper_semilattice) sup_unique: |
22454 | 277 |
fixes f (infixl "\<nabla>" 70) |
25062 | 278 |
assumes ge1 [simp]: "\<And>x y. x \<le> x \<nabla> y" and ge2: "\<And>x y. y \<le> x \<nabla> y" |
279 |
and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<le> x" |
|
22737 | 280 |
shows "x \<squnion> y = x \<nabla> y" |
22454 | 281 |
proof (rule antisym) |
25062 | 282 |
show "x \<squnion> y \<le> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2) |
22454 | 283 |
next |
25062 | 284 |
have leI: "\<And>x y z. x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z" by (blast intro: least) |
285 |
show "x \<nabla> y \<le> x \<squnion> y" by (rule leI) simp_all |
|
22454 | 286 |
qed |
287 |
||
288 |
||
22916 | 289 |
subsection {* @{const min}/@{const max} on linear orders as |
290 |
special case of @{const inf}/@{const sup} *} |
|
291 |
||
292 |
lemma (in linorder) distrib_lattice_min_max: |
|
25062 | 293 |
"distrib_lattice (op \<le>) (op <) min max" |
28823 | 294 |
proof |
25062 | 295 |
have aux: "\<And>x y \<Colon> 'a. x < y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
22916 | 296 |
by (auto simp add: less_le antisym) |
297 |
fix x y z |
|
298 |
show "max x (min y z) = min (max x y) (max x z)" |
|
299 |
unfolding min_def max_def |
|
24640
85a6c200ecd3
Simplified proofs due to transitivity reasoner setup.
ballarin
parents:
24514
diff
changeset
|
300 |
by auto |
22916 | 301 |
qed (auto simp add: min_def max_def not_le less_imp_le) |
21249 | 302 |
|
29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29223
diff
changeset
|
303 |
interpretation min_max!: distrib_lattice "op \<le> :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max |
23948 | 304 |
by (rule distrib_lattice_min_max) |
21249 | 305 |
|
22454 | 306 |
lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
307 |
by (rule ext)+ (auto intro: antisym) |
21733 | 308 |
|
22454 | 309 |
lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
310 |
by (rule ext)+ (auto intro: antisym) |
21733 | 311 |
|
21249 | 312 |
lemmas le_maxI1 = min_max.sup_ge1 |
313 |
lemmas le_maxI2 = min_max.sup_ge2 |
|
21381 | 314 |
|
21249 | 315 |
lemmas max_ac = min_max.sup_assoc min_max.sup_commute |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
316 |
mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute] |
21249 | 317 |
|
318 |
lemmas min_ac = min_max.inf_assoc min_max.inf_commute |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
319 |
mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute] |
21249 | 320 |
|
22454 | 321 |
text {* |
322 |
Now we have inherited antisymmetry as an intro-rule on all |
|
323 |
linear orders. This is a problem because it applies to bool, which is |
|
324 |
undesirable. |
|
325 |
*} |
|
326 |
||
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
327 |
lemmas [rule del] = min_max.le_infI min_max.le_supI |
22454 | 328 |
min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2 |
329 |
min_max.le_infI1 min_max.le_infI2 |
|
330 |
||
331 |
||
23878 | 332 |
subsection {* Complete lattices *} |
333 |
||
28692 | 334 |
class complete_lattice = lattice + bot + top + |
23878 | 335 |
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) |
24345 | 336 |
and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) |
23878 | 337 |
assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" |
24345 | 338 |
and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" |
339 |
assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" |
|
340 |
and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" |
|
23878 | 341 |
begin |
342 |
||
25062 | 343 |
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
344 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
23878 | 345 |
|
25062 | 346 |
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
347 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
23878 | 348 |
|
349 |
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}" |
|
24345 | 350 |
unfolding Sup_Inf by auto |
23878 | 351 |
|
352 |
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}" |
|
353 |
unfolding Inf_Sup by auto |
|
354 |
||
355 |
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" |
|
26233 | 356 |
by (auto intro: antisym Inf_greatest Inf_lower) |
23878 | 357 |
|
24345 | 358 |
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" |
26233 | 359 |
by (auto intro: antisym Sup_least Sup_upper) |
23878 | 360 |
|
361 |
lemma Inf_singleton [simp]: |
|
362 |
"\<Sqinter>{a} = a" |
|
363 |
by (auto intro: antisym Inf_lower Inf_greatest) |
|
364 |
||
24345 | 365 |
lemma Sup_singleton [simp]: |
23878 | 366 |
"\<Squnion>{a} = a" |
367 |
by (auto intro: antisym Sup_upper Sup_least) |
|
368 |
||
369 |
lemma Inf_insert_simp: |
|
370 |
"\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)" |
|
371 |
by (cases "A = {}") (simp_all, simp add: Inf_insert) |
|
372 |
||
373 |
lemma Sup_insert_simp: |
|
374 |
"\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)" |
|
375 |
by (cases "A = {}") (simp_all, simp add: Sup_insert) |
|
376 |
||
377 |
lemma Inf_binary: |
|
378 |
"\<Sqinter>{a, b} = a \<sqinter> b" |
|
379 |
by (simp add: Inf_insert_simp) |
|
380 |
||
381 |
lemma Sup_binary: |
|
382 |
"\<Squnion>{a, b} = a \<squnion> b" |
|
383 |
by (simp add: Sup_insert_simp) |
|
384 |
||
28685 | 385 |
lemma bot_def: |
25206 | 386 |
"bot = \<Squnion>{}" |
28685 | 387 |
by (auto intro: antisym Sup_least) |
23878 | 388 |
|
28692 | 389 |
lemma top_def: |
390 |
"top = \<Sqinter>{}" |
|
391 |
by (auto intro: antisym Inf_greatest) |
|
392 |
||
393 |
lemma sup_bot [simp]: |
|
394 |
"x \<squnion> bot = x" |
|
395 |
using bot_least [of x] by (simp add: le_iff_sup sup_commute) |
|
396 |
||
397 |
lemma inf_top [simp]: |
|
398 |
"x \<sqinter> top = x" |
|
399 |
using top_greatest [of x] by (simp add: le_iff_inf inf_commute) |
|
400 |
||
401 |
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where |
|
25206 | 402 |
"SUPR A f == \<Squnion> (f ` A)" |
23878 | 403 |
|
28692 | 404 |
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where |
25206 | 405 |
"INFI A f == \<Sqinter> (f ` A)" |
23878 | 406 |
|
24749 | 407 |
end |
408 |
||
23878 | 409 |
syntax |
410 |
"_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10) |
|
411 |
"_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10) |
|
412 |
"_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10) |
|
413 |
"_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10) |
|
414 |
||
415 |
translations |
|
416 |
"SUP x y. B" == "SUP x. SUP y. B" |
|
417 |
"SUP x. B" == "CONST SUPR UNIV (%x. B)" |
|
418 |
"SUP x. B" == "SUP x:UNIV. B" |
|
419 |
"SUP x:A. B" == "CONST SUPR A (%x. B)" |
|
420 |
"INF x y. B" == "INF x. INF y. B" |
|
421 |
"INF x. B" == "CONST INFI UNIV (%x. B)" |
|
422 |
"INF x. B" == "INF x:UNIV. B" |
|
423 |
"INF x:A. B" == "CONST INFI A (%x. B)" |
|
424 |
||
425 |
(* To avoid eta-contraction of body: *) |
|
426 |
print_translation {* |
|
427 |
let |
|
428 |
fun btr' syn (A :: Abs abs :: ts) = |
|
429 |
let val (x,t) = atomic_abs_tr' abs |
|
430 |
in list_comb (Syntax.const syn $ x $ A $ t, ts) end |
|
431 |
val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const |
|
432 |
in |
|
433 |
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")] |
|
434 |
end |
|
435 |
*} |
|
436 |
||
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
437 |
context complete_lattice |
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
438 |
begin |
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
439 |
|
23878 | 440 |
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)" |
441 |
by (auto simp add: SUPR_def intro: Sup_upper) |
|
442 |
||
443 |
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u" |
|
444 |
by (auto simp add: SUPR_def intro: Sup_least) |
|
445 |
||
446 |
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i" |
|
447 |
by (auto simp add: INFI_def intro: Inf_lower) |
|
448 |
||
449 |
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)" |
|
450 |
by (auto simp add: INFI_def intro: Inf_greatest) |
|
451 |
||
452 |
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
453 |
by (auto intro: antisym SUP_leI le_SUPI) |
23878 | 454 |
|
455 |
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M" |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
456 |
by (auto intro: antisym INF_leI le_INFI) |
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
457 |
|
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
458 |
end |
23878 | 459 |
|
460 |
||
22454 | 461 |
subsection {* Bool as lattice *} |
462 |
||
25510 | 463 |
instantiation bool :: distrib_lattice |
464 |
begin |
|
465 |
||
466 |
definition |
|
467 |
inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q" |
|
468 |
||
469 |
definition |
|
470 |
sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q" |
|
471 |
||
472 |
instance |
|
22454 | 473 |
by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def) |
474 |
||
25510 | 475 |
end |
476 |
||
477 |
instantiation bool :: complete_lattice |
|
478 |
begin |
|
479 |
||
480 |
definition |
|
481 |
Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)" |
|
482 |
||
483 |
definition |
|
484 |
Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)" |
|
485 |
||
486 |
instance |
|
24345 | 487 |
by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def) |
23878 | 488 |
|
25510 | 489 |
end |
490 |
||
23878 | 491 |
lemma Inf_empty_bool [simp]: |
25206 | 492 |
"\<Sqinter>{}" |
23878 | 493 |
unfolding Inf_bool_def by auto |
494 |
||
495 |
lemma not_Sup_empty_bool [simp]: |
|
496 |
"\<not> Sup {}" |
|
24345 | 497 |
unfolding Sup_bool_def by auto |
23878 | 498 |
|
499 |
||
500 |
subsection {* Fun as lattice *} |
|
501 |
||
25510 | 502 |
instantiation "fun" :: (type, lattice) lattice |
503 |
begin |
|
504 |
||
505 |
definition |
|
28562 | 506 |
inf_fun_eq [code del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)" |
25510 | 507 |
|
508 |
definition |
|
28562 | 509 |
sup_fun_eq [code del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)" |
25510 | 510 |
|
511 |
instance |
|
23878 | 512 |
apply intro_classes |
513 |
unfolding inf_fun_eq sup_fun_eq |
|
514 |
apply (auto intro: le_funI) |
|
515 |
apply (rule le_funI) |
|
516 |
apply (auto dest: le_funD) |
|
517 |
apply (rule le_funI) |
|
518 |
apply (auto dest: le_funD) |
|
519 |
done |
|
520 |
||
25510 | 521 |
end |
23878 | 522 |
|
523 |
instance "fun" :: (type, distrib_lattice) distrib_lattice |
|
524 |
by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1) |
|
525 |
||
25510 | 526 |
instantiation "fun" :: (type, complete_lattice) complete_lattice |
527 |
begin |
|
528 |
||
529 |
definition |
|
28562 | 530 |
Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})" |
25510 | 531 |
|
532 |
definition |
|
28562 | 533 |
Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})" |
25510 | 534 |
|
535 |
instance |
|
24345 | 536 |
by intro_classes |
537 |
(auto simp add: Inf_fun_def Sup_fun_def le_fun_def |
|
538 |
intro: Inf_lower Sup_upper Inf_greatest Sup_least) |
|
23878 | 539 |
|
25510 | 540 |
end |
23878 | 541 |
|
542 |
lemma Inf_empty_fun: |
|
25206 | 543 |
"\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})" |
23878 | 544 |
by rule (auto simp add: Inf_fun_def) |
545 |
||
546 |
lemma Sup_empty_fun: |
|
25206 | 547 |
"\<Squnion>{} = (\<lambda>_. \<Squnion>{})" |
24345 | 548 |
by rule (auto simp add: Sup_fun_def) |
23878 | 549 |
|
550 |
||
26794 | 551 |
subsection {* Set as lattice *} |
552 |
||
553 |
lemma inf_set_eq: "A \<sqinter> B = A \<inter> B" |
|
554 |
apply (rule subset_antisym) |
|
555 |
apply (rule Int_greatest) |
|
556 |
apply (rule inf_le1) |
|
557 |
apply (rule inf_le2) |
|
558 |
apply (rule inf_greatest) |
|
559 |
apply (rule Int_lower1) |
|
560 |
apply (rule Int_lower2) |
|
561 |
done |
|
562 |
||
563 |
lemma sup_set_eq: "A \<squnion> B = A \<union> B" |
|
564 |
apply (rule subset_antisym) |
|
565 |
apply (rule sup_least) |
|
566 |
apply (rule Un_upper1) |
|
567 |
apply (rule Un_upper2) |
|
568 |
apply (rule Un_least) |
|
569 |
apply (rule sup_ge1) |
|
570 |
apply (rule sup_ge2) |
|
571 |
done |
|
572 |
||
573 |
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B" |
|
574 |
apply (fold inf_set_eq sup_set_eq) |
|
575 |
apply (erule mono_inf) |
|
576 |
done |
|
577 |
||
578 |
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)" |
|
579 |
apply (fold inf_set_eq sup_set_eq) |
|
580 |
apply (erule mono_sup) |
|
581 |
done |
|
582 |
||
583 |
lemma Inf_set_eq: "\<Sqinter>S = \<Inter>S" |
|
584 |
apply (rule subset_antisym) |
|
585 |
apply (rule Inter_greatest) |
|
586 |
apply (erule Inf_lower) |
|
587 |
apply (rule Inf_greatest) |
|
588 |
apply (erule Inter_lower) |
|
589 |
done |
|
590 |
||
591 |
lemma Sup_set_eq: "\<Squnion>S = \<Union>S" |
|
592 |
apply (rule subset_antisym) |
|
593 |
apply (rule Sup_least) |
|
594 |
apply (erule Union_upper) |
|
595 |
apply (rule Union_least) |
|
596 |
apply (erule Sup_upper) |
|
597 |
done |
|
598 |
||
599 |
lemma top_set_eq: "top = UNIV" |
|
600 |
by (iprover intro!: subset_antisym subset_UNIV top_greatest) |
|
601 |
||
602 |
lemma bot_set_eq: "bot = {}" |
|
603 |
by (iprover intro!: subset_antisym empty_subsetI bot_least) |
|
604 |
||
605 |
||
23878 | 606 |
text {* redundant bindings *} |
22454 | 607 |
|
608 |
lemmas inf_aci = inf_ACI |
|
609 |
lemmas sup_aci = sup_ACI |
|
610 |
||
25062 | 611 |
no_notation |
25382 | 612 |
less_eq (infix "\<sqsubseteq>" 50) and |
613 |
less (infix "\<sqsubset>" 50) and |
|
614 |
inf (infixl "\<sqinter>" 70) and |
|
615 |
sup (infixl "\<squnion>" 65) and |
|
616 |
Inf ("\<Sqinter>_" [900] 900) and |
|
617 |
Sup ("\<Squnion>_" [900] 900) |
|
25062 | 618 |
|
21249 | 619 |
end |