author | wenzelm |
Thu, 12 Aug 2010 13:49:08 +0200 | |
changeset 38360 | 53224a4d2f0e |
parent 37770 | cddb3106adb8 |
child 39970 | 9023b897e67a |
permissions | -rw-r--r-- |
25904 | 1 |
(* Title: HOLCF/LowerPD.thy |
2 |
Author: Brian Huffman |
|
3 |
*) |
|
4 |
||
5 |
header {* Lower powerdomain *} |
|
6 |
||
7 |
theory LowerPD |
|
8 |
imports CompactBasis |
|
9 |
begin |
|
10 |
||
11 |
subsection {* Basis preorder *} |
|
12 |
||
13 |
definition |
|
14 |
lower_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<flat>" 50) where |
|
26420
57a626f64875
make preorder locale into a superclass of class po
huffman
parents:
26407
diff
changeset
|
15 |
"lower_le = (\<lambda>u v. \<forall>x\<in>Rep_pd_basis u. \<exists>y\<in>Rep_pd_basis v. x \<sqsubseteq> y)" |
25904 | 16 |
|
17 |
lemma lower_le_refl [simp]: "t \<le>\<flat> t" |
|
26420
57a626f64875
make preorder locale into a superclass of class po
huffman
parents:
26407
diff
changeset
|
18 |
unfolding lower_le_def by fast |
25904 | 19 |
|
20 |
lemma lower_le_trans: "\<lbrakk>t \<le>\<flat> u; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> t \<le>\<flat> v" |
|
21 |
unfolding lower_le_def |
|
22 |
apply (rule ballI) |
|
23 |
apply (drule (1) bspec, erule bexE) |
|
24 |
apply (drule (1) bspec, erule bexE) |
|
25 |
apply (erule rev_bexI) |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
26 |
apply (erule (1) below_trans) |
25904 | 27 |
done |
28 |
||
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29990
diff
changeset
|
29 |
interpretation lower_le: preorder lower_le |
25904 | 30 |
by (rule preorder.intro, rule lower_le_refl, rule lower_le_trans) |
31 |
||
32 |
lemma lower_le_minimal [simp]: "PDUnit compact_bot \<le>\<flat> t" |
|
33 |
unfolding lower_le_def Rep_PDUnit |
|
34 |
by (simp, rule Rep_pd_basis_nonempty [folded ex_in_conv]) |
|
35 |
||
26420
57a626f64875
make preorder locale into a superclass of class po
huffman
parents:
26407
diff
changeset
|
36 |
lemma PDUnit_lower_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<flat> PDUnit y" |
25904 | 37 |
unfolding lower_le_def Rep_PDUnit by fast |
38 |
||
39 |
lemma PDPlus_lower_mono: "\<lbrakk>s \<le>\<flat> t; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<flat> PDPlus t v" |
|
40 |
unfolding lower_le_def Rep_PDPlus by fast |
|
41 |
||
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
42 |
lemma PDPlus_lower_le: "t \<le>\<flat> PDPlus t u" |
26420
57a626f64875
make preorder locale into a superclass of class po
huffman
parents:
26407
diff
changeset
|
43 |
unfolding lower_le_def Rep_PDPlus by fast |
25904 | 44 |
|
45 |
lemma lower_le_PDUnit_PDUnit_iff [simp]: |
|
26420
57a626f64875
make preorder locale into a superclass of class po
huffman
parents:
26407
diff
changeset
|
46 |
"(PDUnit a \<le>\<flat> PDUnit b) = a \<sqsubseteq> b" |
25904 | 47 |
unfolding lower_le_def Rep_PDUnit by fast |
48 |
||
49 |
lemma lower_le_PDUnit_PDPlus_iff: |
|
50 |
"(PDUnit a \<le>\<flat> PDPlus t u) = (PDUnit a \<le>\<flat> t \<or> PDUnit a \<le>\<flat> u)" |
|
51 |
unfolding lower_le_def Rep_PDPlus Rep_PDUnit by fast |
|
52 |
||
53 |
lemma lower_le_PDPlus_iff: "(PDPlus t u \<le>\<flat> v) = (t \<le>\<flat> v \<and> u \<le>\<flat> v)" |
|
54 |
unfolding lower_le_def Rep_PDPlus by fast |
|
55 |
||
56 |
lemma lower_le_induct [induct set: lower_le]: |
|
57 |
assumes le: "t \<le>\<flat> u" |
|
26420
57a626f64875
make preorder locale into a superclass of class po
huffman
parents:
26407
diff
changeset
|
58 |
assumes 1: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)" |
25904 | 59 |
assumes 2: "\<And>t u a. P (PDUnit a) t \<Longrightarrow> P (PDUnit a) (PDPlus t u)" |
60 |
assumes 3: "\<And>t u v. \<lbrakk>P t v; P u v\<rbrakk> \<Longrightarrow> P (PDPlus t u) v" |
|
61 |
shows "P t u" |
|
62 |
using le |
|
63 |
apply (induct t arbitrary: u rule: pd_basis_induct) |
|
64 |
apply (erule rev_mp) |
|
65 |
apply (induct_tac u rule: pd_basis_induct) |
|
66 |
apply (simp add: 1) |
|
67 |
apply (simp add: lower_le_PDUnit_PDPlus_iff) |
|
68 |
apply (simp add: 2) |
|
69 |
apply (subst PDPlus_commute) |
|
70 |
apply (simp add: 2) |
|
71 |
apply (simp add: lower_le_PDPlus_iff 3) |
|
72 |
done |
|
73 |
||
27405 | 74 |
lemma pd_take_lower_chain: |
75 |
"pd_take n t \<le>\<flat> pd_take (Suc n) t" |
|
25904 | 76 |
apply (induct t rule: pd_basis_induct) |
27289 | 77 |
apply (simp add: compact_basis.take_chain) |
25904 | 78 |
apply (simp add: PDPlus_lower_mono) |
79 |
done |
|
80 |
||
27405 | 81 |
lemma pd_take_lower_le: "pd_take i t \<le>\<flat> t" |
25904 | 82 |
apply (induct t rule: pd_basis_induct) |
27289 | 83 |
apply (simp add: compact_basis.take_less) |
25904 | 84 |
apply (simp add: PDPlus_lower_mono) |
85 |
done |
|
86 |
||
27405 | 87 |
lemma pd_take_lower_mono: |
88 |
"t \<le>\<flat> u \<Longrightarrow> pd_take n t \<le>\<flat> pd_take n u" |
|
25904 | 89 |
apply (erule lower_le_induct) |
27289 | 90 |
apply (simp add: compact_basis.take_mono) |
25904 | 91 |
apply (simp add: lower_le_PDUnit_PDPlus_iff) |
92 |
apply (simp add: lower_le_PDPlus_iff) |
|
93 |
done |
|
94 |
||
95 |
||
96 |
subsection {* Type definition *} |
|
97 |
||
27373
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
98 |
typedef (open) 'a lower_pd = |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
99 |
"{S::'a pd_basis set. lower_le.ideal S}" |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
100 |
by (fast intro: lower_le.ideal_principal) |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
101 |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
102 |
instantiation lower_pd :: (profinite) below |
27373
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
103 |
begin |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
104 |
|
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
105 |
definition |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
106 |
"x \<sqsubseteq> y \<longleftrightarrow> Rep_lower_pd x \<subseteq> Rep_lower_pd y" |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
107 |
|
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
108 |
instance .. |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
109 |
end |
25904 | 110 |
|
27373
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
111 |
instance lower_pd :: (profinite) po |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
112 |
by (rule lower_le.typedef_ideal_po |
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
113 |
[OF type_definition_lower_pd below_lower_pd_def]) |
27373
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
114 |
|
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
115 |
instance lower_pd :: (profinite) cpo |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
116 |
by (rule lower_le.typedef_ideal_cpo |
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
117 |
[OF type_definition_lower_pd below_lower_pd_def]) |
27373
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
118 |
|
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
119 |
lemma Rep_lower_pd_lub: |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
120 |
"chain Y \<Longrightarrow> Rep_lower_pd (\<Squnion>i. Y i) = (\<Union>i. Rep_lower_pd (Y i))" |
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
121 |
by (rule lower_le.typedef_ideal_rep_contlub |
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
122 |
[OF type_definition_lower_pd below_lower_pd_def]) |
27373
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
123 |
|
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
124 |
lemma ideal_Rep_lower_pd: "lower_le.ideal (Rep_lower_pd xs)" |
26927 | 125 |
by (rule Rep_lower_pd [unfolded mem_Collect_eq]) |
25904 | 126 |
|
127 |
definition |
|
128 |
lower_principal :: "'a pd_basis \<Rightarrow> 'a lower_pd" where |
|
27373
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
129 |
"lower_principal t = Abs_lower_pd {u. u \<le>\<flat> t}" |
25904 | 130 |
|
131 |
lemma Rep_lower_principal: |
|
27373
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
132 |
"Rep_lower_pd (lower_principal t) = {u. u \<le>\<flat> t}" |
25904 | 133 |
unfolding lower_principal_def |
27297
2c42b1505f25
removed SetPcpo.thy and cpo instance for type bool;
huffman
parents:
27289
diff
changeset
|
134 |
by (simp add: Abs_lower_pd_inverse lower_le.ideal_principal) |
25904 | 135 |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29990
diff
changeset
|
136 |
interpretation lower_pd: |
29237 | 137 |
ideal_completion lower_le pd_take lower_principal Rep_lower_pd |
25904 | 138 |
apply unfold_locales |
27405 | 139 |
apply (rule pd_take_lower_le) |
140 |
apply (rule pd_take_idem) |
|
141 |
apply (erule pd_take_lower_mono) |
|
142 |
apply (rule pd_take_lower_chain) |
|
143 |
apply (rule finite_range_pd_take) |
|
144 |
apply (rule pd_take_covers) |
|
26420
57a626f64875
make preorder locale into a superclass of class po
huffman
parents:
26407
diff
changeset
|
145 |
apply (rule ideal_Rep_lower_pd) |
27373
5794a0e3e26c
remove cset theory; define ideal completions using typedef instead of cpodef
huffman
parents:
27310
diff
changeset
|
146 |
apply (erule Rep_lower_pd_lub) |
26420
57a626f64875
make preorder locale into a superclass of class po
huffman
parents:
26407
diff
changeset
|
147 |
apply (rule Rep_lower_principal) |
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
148 |
apply (simp only: below_lower_pd_def) |
25904 | 149 |
done |
150 |
||
27289 | 151 |
text {* Lower powerdomain is pointed *} |
25904 | 152 |
|
153 |
lemma lower_pd_minimal: "lower_principal (PDUnit compact_bot) \<sqsubseteq> ys" |
|
154 |
by (induct ys rule: lower_pd.principal_induct, simp, simp) |
|
155 |
||
156 |
instance lower_pd :: (bifinite) pcpo |
|
26927 | 157 |
by intro_classes (fast intro: lower_pd_minimal) |
25904 | 158 |
|
159 |
lemma inst_lower_pd_pcpo: "\<bottom> = lower_principal (PDUnit compact_bot)" |
|
160 |
by (rule lower_pd_minimal [THEN UU_I, symmetric]) |
|
161 |
||
27289 | 162 |
text {* Lower powerdomain is profinite *} |
25904 | 163 |
|
26962
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26927
diff
changeset
|
164 |
instantiation lower_pd :: (profinite) profinite |
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26927
diff
changeset
|
165 |
begin |
25904 | 166 |
|
26962
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26927
diff
changeset
|
167 |
definition |
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26927
diff
changeset
|
168 |
approx_lower_pd_def: "approx = lower_pd.completion_approx" |
26927 | 169 |
|
26962
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26927
diff
changeset
|
170 |
instance |
26927 | 171 |
apply (intro_classes, unfold approx_lower_pd_def) |
27310 | 172 |
apply (rule lower_pd.chain_completion_approx) |
26927 | 173 |
apply (rule lower_pd.lub_completion_approx) |
174 |
apply (rule lower_pd.completion_approx_idem) |
|
175 |
apply (rule lower_pd.finite_fixes_completion_approx) |
|
176 |
done |
|
177 |
||
26962
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26927
diff
changeset
|
178 |
end |
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
26927
diff
changeset
|
179 |
|
26927 | 180 |
instance lower_pd :: (bifinite) bifinite .. |
25904 | 181 |
|
182 |
lemma approx_lower_principal [simp]: |
|
27405 | 183 |
"approx n\<cdot>(lower_principal t) = lower_principal (pd_take n t)" |
25904 | 184 |
unfolding approx_lower_pd_def |
26927 | 185 |
by (rule lower_pd.completion_approx_principal) |
25904 | 186 |
|
187 |
lemma approx_eq_lower_principal: |
|
27405 | 188 |
"\<exists>t\<in>Rep_lower_pd xs. approx n\<cdot>xs = lower_principal (pd_take n t)" |
25904 | 189 |
unfolding approx_lower_pd_def |
26927 | 190 |
by (rule lower_pd.completion_approx_eq_principal) |
26407
562a1d615336
rename class bifinite_cpo to profinite; generalize powerdomains from bifinite to profinite
huffman
parents:
26041
diff
changeset
|
191 |
|
25904 | 192 |
|
26927 | 193 |
subsection {* Monadic unit and plus *} |
25904 | 194 |
|
195 |
definition |
|
196 |
lower_unit :: "'a \<rightarrow> 'a lower_pd" where |
|
197 |
"lower_unit = compact_basis.basis_fun (\<lambda>a. lower_principal (PDUnit a))" |
|
198 |
||
199 |
definition |
|
200 |
lower_plus :: "'a lower_pd \<rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd" where |
|
201 |
"lower_plus = lower_pd.basis_fun (\<lambda>t. lower_pd.basis_fun (\<lambda>u. |
|
202 |
lower_principal (PDPlus t u)))" |
|
203 |
||
204 |
abbreviation |
|
205 |
lower_add :: "'a lower_pd \<Rightarrow> 'a lower_pd \<Rightarrow> 'a lower_pd" |
|
206 |
(infixl "+\<flat>" 65) where |
|
207 |
"xs +\<flat> ys == lower_plus\<cdot>xs\<cdot>ys" |
|
208 |
||
26927 | 209 |
syntax |
210 |
"_lower_pd" :: "args \<Rightarrow> 'a lower_pd" ("{_}\<flat>") |
|
211 |
||
212 |
translations |
|
213 |
"{x,xs}\<flat>" == "{x}\<flat> +\<flat> {xs}\<flat>" |
|
214 |
"{x}\<flat>" == "CONST lower_unit\<cdot>x" |
|
215 |
||
216 |
lemma lower_unit_Rep_compact_basis [simp]: |
|
217 |
"{Rep_compact_basis a}\<flat> = lower_principal (PDUnit a)" |
|
218 |
unfolding lower_unit_def |
|
27289 | 219 |
by (simp add: compact_basis.basis_fun_principal PDUnit_lower_mono) |
26927 | 220 |
|
25904 | 221 |
lemma lower_plus_principal [simp]: |
26927 | 222 |
"lower_principal t +\<flat> lower_principal u = lower_principal (PDPlus t u)" |
25904 | 223 |
unfolding lower_plus_def |
224 |
by (simp add: lower_pd.basis_fun_principal |
|
225 |
lower_pd.basis_fun_mono PDPlus_lower_mono) |
|
226 |
||
26927 | 227 |
lemma approx_lower_unit [simp]: |
228 |
"approx n\<cdot>{x}\<flat> = {approx n\<cdot>x}\<flat>" |
|
27289 | 229 |
apply (induct x rule: compact_basis.principal_induct, simp) |
26927 | 230 |
apply (simp add: approx_Rep_compact_basis) |
231 |
done |
|
232 |
||
25904 | 233 |
lemma approx_lower_plus [simp]: |
26927 | 234 |
"approx n\<cdot>(xs +\<flat> ys) = (approx n\<cdot>xs) +\<flat> (approx n\<cdot>ys)" |
27289 | 235 |
by (induct xs ys rule: lower_pd.principal_induct2, simp, simp, simp) |
25904 | 236 |
|
37770
cddb3106adb8
avoid explicit mandatory prefix markers when prefixes are mandatory implicitly
haftmann
parents:
36635
diff
changeset
|
237 |
interpretation lower_add: semilattice lower_add proof |
34973
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
238 |
fix xs ys zs :: "'a lower_pd" |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
239 |
show "(xs +\<flat> ys) +\<flat> zs = xs +\<flat> (ys +\<flat> zs)" |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
240 |
apply (induct xs ys arbitrary: zs rule: lower_pd.principal_induct2, simp, simp) |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
241 |
apply (rule_tac x=zs in lower_pd.principal_induct, simp) |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
242 |
apply (simp add: PDPlus_assoc) |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
243 |
done |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
244 |
show "xs +\<flat> ys = ys +\<flat> xs" |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
245 |
apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp) |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
246 |
apply (simp add: PDPlus_commute) |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
247 |
done |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
248 |
show "xs +\<flat> xs = xs" |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
249 |
apply (induct xs rule: lower_pd.principal_induct, simp) |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
250 |
apply (simp add: PDPlus_absorb) |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
251 |
done |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
252 |
qed |
26927 | 253 |
|
34973
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
254 |
lemmas lower_plus_assoc = lower_add.assoc |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
255 |
lemmas lower_plus_commute = lower_add.commute |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
256 |
lemmas lower_plus_absorb = lower_add.idem |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
257 |
lemmas lower_plus_left_commute = lower_add.left_commute |
ae634fad947e
dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents:
33808
diff
changeset
|
258 |
lemmas lower_plus_left_absorb = lower_add.left_idem |
26927 | 259 |
|
29990
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
260 |
text {* Useful for @{text "simp add: lower_plus_ac"} *} |
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
261 |
lemmas lower_plus_ac = |
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
262 |
lower_plus_assoc lower_plus_commute lower_plus_left_commute |
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
263 |
|
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
264 |
text {* Useful for @{text "simp only: lower_plus_aci"} *} |
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
265 |
lemmas lower_plus_aci = |
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
266 |
lower_plus_ac lower_plus_absorb lower_plus_left_absorb |
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
267 |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
268 |
lemma lower_plus_below1: "xs \<sqsubseteq> xs +\<flat> ys" |
27289 | 269 |
apply (induct xs ys rule: lower_pd.principal_induct2, simp, simp) |
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
270 |
apply (simp add: PDPlus_lower_le) |
25904 | 271 |
done |
272 |
||
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
273 |
lemma lower_plus_below2: "ys \<sqsubseteq> xs +\<flat> ys" |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
274 |
by (subst lower_plus_commute, rule lower_plus_below1) |
25904 | 275 |
|
26927 | 276 |
lemma lower_plus_least: "\<lbrakk>xs \<sqsubseteq> zs; ys \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> xs +\<flat> ys \<sqsubseteq> zs" |
25904 | 277 |
apply (subst lower_plus_absorb [of zs, symmetric]) |
278 |
apply (erule (1) monofun_cfun [OF monofun_cfun_arg]) |
|
279 |
done |
|
280 |
||
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
281 |
lemma lower_plus_below_iff: |
26927 | 282 |
"xs +\<flat> ys \<sqsubseteq> zs \<longleftrightarrow> xs \<sqsubseteq> zs \<and> ys \<sqsubseteq> zs" |
25904 | 283 |
apply safe |
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
284 |
apply (erule below_trans [OF lower_plus_below1]) |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
285 |
apply (erule below_trans [OF lower_plus_below2]) |
25904 | 286 |
apply (erule (1) lower_plus_least) |
287 |
done |
|
288 |
||
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
289 |
lemma lower_unit_below_plus_iff: |
26927 | 290 |
"{x}\<flat> \<sqsubseteq> ys +\<flat> zs \<longleftrightarrow> {x}\<flat> \<sqsubseteq> ys \<or> {x}\<flat> \<sqsubseteq> zs" |
25904 | 291 |
apply (rule iffI) |
292 |
apply (subgoal_tac |
|
26927 | 293 |
"adm (\<lambda>f. f\<cdot>{x}\<flat> \<sqsubseteq> f\<cdot>ys \<or> f\<cdot>{x}\<flat> \<sqsubseteq> f\<cdot>zs)") |
25925 | 294 |
apply (drule admD, rule chain_approx) |
25904 | 295 |
apply (drule_tac f="approx i" in monofun_cfun_arg) |
27289 | 296 |
apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp) |
297 |
apply (cut_tac x="approx i\<cdot>ys" in lower_pd.compact_imp_principal, simp) |
|
298 |
apply (cut_tac x="approx i\<cdot>zs" in lower_pd.compact_imp_principal, simp) |
|
25904 | 299 |
apply (clarify, simp add: lower_le_PDUnit_PDPlus_iff) |
300 |
apply simp |
|
301 |
apply simp |
|
302 |
apply (erule disjE) |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
303 |
apply (erule below_trans [OF _ lower_plus_below1]) |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
304 |
apply (erule below_trans [OF _ lower_plus_below2]) |
25904 | 305 |
done |
306 |
||
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
307 |
lemma lower_unit_below_iff [simp]: "{x}\<flat> \<sqsubseteq> {y}\<flat> \<longleftrightarrow> x \<sqsubseteq> y" |
26927 | 308 |
apply (rule iffI) |
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
309 |
apply (rule profinite_below_ext) |
26927 | 310 |
apply (drule_tac f="approx i" in monofun_cfun_arg, simp) |
27289 | 311 |
apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp) |
312 |
apply (cut_tac x="approx i\<cdot>y" in compact_basis.compact_imp_principal, simp) |
|
313 |
apply clarsimp |
|
26927 | 314 |
apply (erule monofun_cfun_arg) |
315 |
done |
|
316 |
||
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
317 |
lemmas lower_pd_below_simps = |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
318 |
lower_unit_below_iff |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
319 |
lower_plus_below_iff |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
320 |
lower_unit_below_plus_iff |
25904 | 321 |
|
26927 | 322 |
lemma lower_unit_eq_iff [simp]: "{x}\<flat> = {y}\<flat> \<longleftrightarrow> x = y" |
29990
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
323 |
by (simp add: po_eq_conv) |
26927 | 324 |
|
325 |
lemma lower_unit_strict [simp]: "{\<bottom>}\<flat> = \<bottom>" |
|
326 |
unfolding inst_lower_pd_pcpo Rep_compact_bot [symmetric] by simp |
|
327 |
||
328 |
lemma lower_unit_strict_iff [simp]: "{x}\<flat> = \<bottom> \<longleftrightarrow> x = \<bottom>" |
|
329 |
unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff) |
|
330 |
||
331 |
lemma lower_plus_strict_iff [simp]: |
|
332 |
"xs +\<flat> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<and> ys = \<bottom>" |
|
333 |
apply safe |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
334 |
apply (rule UU_I, erule subst, rule lower_plus_below1) |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
335 |
apply (rule UU_I, erule subst, rule lower_plus_below2) |
26927 | 336 |
apply (rule lower_plus_absorb) |
337 |
done |
|
338 |
||
339 |
lemma lower_plus_strict1 [simp]: "\<bottom> +\<flat> ys = ys" |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
340 |
apply (rule below_antisym [OF _ lower_plus_below2]) |
26927 | 341 |
apply (simp add: lower_plus_least) |
342 |
done |
|
343 |
||
344 |
lemma lower_plus_strict2 [simp]: "xs +\<flat> \<bottom> = xs" |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
345 |
apply (rule below_antisym [OF _ lower_plus_below1]) |
26927 | 346 |
apply (simp add: lower_plus_least) |
347 |
done |
|
348 |
||
349 |
lemma compact_lower_unit_iff [simp]: "compact {x}\<flat> \<longleftrightarrow> compact x" |
|
27309 | 350 |
unfolding profinite_compact_iff by simp |
26927 | 351 |
|
352 |
lemma compact_lower_plus [simp]: |
|
353 |
"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<flat> ys)" |
|
27289 | 354 |
by (auto dest!: lower_pd.compact_imp_principal) |
26927 | 355 |
|
25904 | 356 |
|
357 |
subsection {* Induction rules *} |
|
358 |
||
359 |
lemma lower_pd_induct1: |
|
360 |
assumes P: "adm P" |
|
26927 | 361 |
assumes unit: "\<And>x. P {x}\<flat>" |
25904 | 362 |
assumes insert: |
26927 | 363 |
"\<And>x ys. \<lbrakk>P {x}\<flat>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<flat> +\<flat> ys)" |
25904 | 364 |
shows "P (xs::'a lower_pd)" |
27289 | 365 |
apply (induct xs rule: lower_pd.principal_induct, rule P) |
366 |
apply (induct_tac a rule: pd_basis_induct1) |
|
25904 | 367 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric]) |
368 |
apply (rule unit) |
|
369 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric] |
|
370 |
lower_plus_principal [symmetric]) |
|
371 |
apply (erule insert [OF unit]) |
|
372 |
done |
|
373 |
||
374 |
lemma lower_pd_induct: |
|
375 |
assumes P: "adm P" |
|
26927 | 376 |
assumes unit: "\<And>x. P {x}\<flat>" |
377 |
assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<flat> ys)" |
|
25904 | 378 |
shows "P (xs::'a lower_pd)" |
27289 | 379 |
apply (induct xs rule: lower_pd.principal_induct, rule P) |
380 |
apply (induct_tac a rule: pd_basis_induct) |
|
25904 | 381 |
apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit) |
382 |
apply (simp only: lower_plus_principal [symmetric] plus) |
|
383 |
done |
|
384 |
||
385 |
||
386 |
subsection {* Monadic bind *} |
|
387 |
||
388 |
definition |
|
389 |
lower_bind_basis :: |
|
390 |
"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where |
|
391 |
"lower_bind_basis = fold_pd |
|
392 |
(\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a)) |
|
26927 | 393 |
(\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)" |
25904 | 394 |
|
26927 | 395 |
lemma ACI_lower_bind: |
36635
080b755377c0
locale predicates of classes carry a mandatory "class" prefix
haftmann
parents:
35901
diff
changeset
|
396 |
"class.ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<flat> y\<cdot>f)" |
25904 | 397 |
apply unfold_locales |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25925
diff
changeset
|
398 |
apply (simp add: lower_plus_assoc) |
25904 | 399 |
apply (simp add: lower_plus_commute) |
29990
b11793ea15a3
avoid using ab_semigroup_idem_mult locale for powerdomains
huffman
parents:
29672
diff
changeset
|
400 |
apply (simp add: eta_cfun) |
25904 | 401 |
done |
402 |
||
403 |
lemma lower_bind_basis_simps [simp]: |
|
404 |
"lower_bind_basis (PDUnit a) = |
|
405 |
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))" |
|
406 |
"lower_bind_basis (PDPlus t u) = |
|
26927 | 407 |
(\<Lambda> f. lower_bind_basis t\<cdot>f +\<flat> lower_bind_basis u\<cdot>f)" |
25904 | 408 |
unfolding lower_bind_basis_def |
409 |
apply - |
|
26927 | 410 |
apply (rule fold_pd_PDUnit [OF ACI_lower_bind]) |
411 |
apply (rule fold_pd_PDPlus [OF ACI_lower_bind]) |
|
25904 | 412 |
done |
413 |
||
414 |
lemma lower_bind_basis_mono: |
|
415 |
"t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u" |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
416 |
unfolding expand_cfun_below |
25904 | 417 |
apply (erule lower_le_induct, safe) |
27289 | 418 |
apply (simp add: monofun_cfun) |
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
419 |
apply (simp add: rev_below_trans [OF lower_plus_below1]) |
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
30729
diff
changeset
|
420 |
apply (simp add: lower_plus_below_iff) |
25904 | 421 |
done |
422 |
||
423 |
definition |
|
424 |
lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where |
|
425 |
"lower_bind = lower_pd.basis_fun lower_bind_basis" |
|
426 |
||
427 |
lemma lower_bind_principal [simp]: |
|
428 |
"lower_bind\<cdot>(lower_principal t) = lower_bind_basis t" |
|
429 |
unfolding lower_bind_def |
|
430 |
apply (rule lower_pd.basis_fun_principal) |
|
431 |
apply (erule lower_bind_basis_mono) |
|
432 |
done |
|
433 |
||
434 |
lemma lower_bind_unit [simp]: |
|
26927 | 435 |
"lower_bind\<cdot>{x}\<flat>\<cdot>f = f\<cdot>x" |
27289 | 436 |
by (induct x rule: compact_basis.principal_induct, simp, simp) |
25904 | 437 |
|
438 |
lemma lower_bind_plus [simp]: |
|
26927 | 439 |
"lower_bind\<cdot>(xs +\<flat> ys)\<cdot>f = lower_bind\<cdot>xs\<cdot>f +\<flat> lower_bind\<cdot>ys\<cdot>f" |
27289 | 440 |
by (induct xs ys rule: lower_pd.principal_induct2, simp, simp, simp) |
25904 | 441 |
|
442 |
lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>" |
|
443 |
unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit) |
|
444 |
||
445 |
||
446 |
subsection {* Map and join *} |
|
447 |
||
448 |
definition |
|
449 |
lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where |
|
26927 | 450 |
"lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<flat>))" |
25904 | 451 |
|
452 |
definition |
|
453 |
lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where |
|
454 |
"lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))" |
|
455 |
||
456 |
lemma lower_map_unit [simp]: |
|
26927 | 457 |
"lower_map\<cdot>f\<cdot>{x}\<flat> = {f\<cdot>x}\<flat>" |
25904 | 458 |
unfolding lower_map_def by simp |
459 |
||
460 |
lemma lower_map_plus [simp]: |
|
26927 | 461 |
"lower_map\<cdot>f\<cdot>(xs +\<flat> ys) = lower_map\<cdot>f\<cdot>xs +\<flat> lower_map\<cdot>f\<cdot>ys" |
25904 | 462 |
unfolding lower_map_def by simp |
463 |
||
464 |
lemma lower_join_unit [simp]: |
|
26927 | 465 |
"lower_join\<cdot>{xs}\<flat> = xs" |
25904 | 466 |
unfolding lower_join_def by simp |
467 |
||
468 |
lemma lower_join_plus [simp]: |
|
26927 | 469 |
"lower_join\<cdot>(xss +\<flat> yss) = lower_join\<cdot>xss +\<flat> lower_join\<cdot>yss" |
25904 | 470 |
unfolding lower_join_def by simp |
471 |
||
472 |
lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs" |
|
473 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
474 |
||
33808 | 475 |
lemma lower_map_ID: "lower_map\<cdot>ID = ID" |
476 |
by (simp add: expand_cfun_eq ID_def lower_map_ident) |
|
477 |
||
25904 | 478 |
lemma lower_map_map: |
479 |
"lower_map\<cdot>f\<cdot>(lower_map\<cdot>g\<cdot>xs) = lower_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs" |
|
480 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
481 |
||
482 |
lemma lower_join_map_unit: |
|
483 |
"lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs" |
|
484 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
485 |
||
486 |
lemma lower_join_map_join: |
|
487 |
"lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)" |
|
488 |
by (induct xsss rule: lower_pd_induct, simp_all) |
|
489 |
||
490 |
lemma lower_join_map_map: |
|
491 |
"lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) = |
|
492 |
lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)" |
|
493 |
by (induct xss rule: lower_pd_induct, simp_all) |
|
494 |
||
495 |
lemma lower_map_approx: "lower_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs" |
|
496 |
by (induct xs rule: lower_pd_induct, simp_all) |
|
497 |
||
33585
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
498 |
lemma ep_pair_lower_map: "ep_pair e p \<Longrightarrow> ep_pair (lower_map\<cdot>e) (lower_map\<cdot>p)" |
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
499 |
apply default |
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
500 |
apply (induct_tac x rule: lower_pd_induct, simp_all add: ep_pair.e_inverse) |
35901
12f09bf2c77f
fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents:
34973
diff
changeset
|
501 |
apply (induct_tac y rule: lower_pd_induct) |
12f09bf2c77f
fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents:
34973
diff
changeset
|
502 |
apply (simp_all add: ep_pair.e_p_below monofun_cfun) |
33585
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
503 |
done |
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
504 |
|
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
505 |
lemma deflation_lower_map: "deflation d \<Longrightarrow> deflation (lower_map\<cdot>d)" |
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
506 |
apply default |
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
507 |
apply (induct_tac x rule: lower_pd_induct, simp_all add: deflation.idem) |
35901
12f09bf2c77f
fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents:
34973
diff
changeset
|
508 |
apply (induct_tac x rule: lower_pd_induct) |
12f09bf2c77f
fix LaTeX overfull hbox warnings in HOLCF document
huffman
parents:
34973
diff
changeset
|
509 |
apply (simp_all add: deflation.below monofun_cfun) |
33585
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
510 |
done |
8d39394fe5cf
ep_pair and deflation lemmas for powerdomain map functions
huffman
parents:
31076
diff
changeset
|
511 |
|
25904 | 512 |
end |