huffman@25904
|
1 |
(* Title: HOLCF/ConvexPD.thy
|
huffman@25904
|
2 |
Author: Brian Huffman
|
huffman@25904
|
3 |
*)
|
huffman@25904
|
4 |
|
huffman@25904
|
5 |
header {* Convex powerdomain *}
|
huffman@25904
|
6 |
|
huffman@25904
|
7 |
theory ConvexPD
|
huffman@25904
|
8 |
imports UpperPD LowerPD
|
huffman@25904
|
9 |
begin
|
huffman@25904
|
10 |
|
huffman@25904
|
11 |
subsection {* Basis preorder *}
|
huffman@25904
|
12 |
|
huffman@25904
|
13 |
definition
|
huffman@25904
|
14 |
convex_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<natural>" 50) where
|
huffman@25904
|
15 |
"convex_le = (\<lambda>u v. u \<le>\<sharp> v \<and> u \<le>\<flat> v)"
|
huffman@25904
|
16 |
|
huffman@25904
|
17 |
lemma convex_le_refl [simp]: "t \<le>\<natural> t"
|
huffman@25904
|
18 |
unfolding convex_le_def by (fast intro: upper_le_refl lower_le_refl)
|
huffman@25904
|
19 |
|
huffman@25904
|
20 |
lemma convex_le_trans: "\<lbrakk>t \<le>\<natural> u; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> t \<le>\<natural> v"
|
huffman@25904
|
21 |
unfolding convex_le_def by (fast intro: upper_le_trans lower_le_trans)
|
huffman@25904
|
22 |
|
huffman@25904
|
23 |
interpretation convex_le: preorder [convex_le]
|
huffman@25904
|
24 |
by (rule preorder.intro, rule convex_le_refl, rule convex_le_trans)
|
huffman@25904
|
25 |
|
huffman@25904
|
26 |
lemma upper_le_minimal [simp]: "PDUnit compact_bot \<le>\<natural> t"
|
huffman@25904
|
27 |
unfolding convex_le_def Rep_PDUnit by simp
|
huffman@25904
|
28 |
|
huffman@26420
|
29 |
lemma PDUnit_convex_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<natural> PDUnit y"
|
huffman@25904
|
30 |
unfolding convex_le_def by (fast intro: PDUnit_upper_mono PDUnit_lower_mono)
|
huffman@25904
|
31 |
|
huffman@25904
|
32 |
lemma PDPlus_convex_mono: "\<lbrakk>s \<le>\<natural> t; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<natural> PDPlus t v"
|
huffman@25904
|
33 |
unfolding convex_le_def by (fast intro: PDPlus_upper_mono PDPlus_lower_mono)
|
huffman@25904
|
34 |
|
huffman@25904
|
35 |
lemma convex_le_PDUnit_PDUnit_iff [simp]:
|
huffman@26420
|
36 |
"(PDUnit a \<le>\<natural> PDUnit b) = a \<sqsubseteq> b"
|
huffman@25904
|
37 |
unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit by fast
|
huffman@25904
|
38 |
|
huffman@25904
|
39 |
lemma convex_le_PDUnit_lemma1:
|
huffman@26420
|
40 |
"(PDUnit a \<le>\<natural> t) = (\<forall>b\<in>Rep_pd_basis t. a \<sqsubseteq> b)"
|
huffman@25904
|
41 |
unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit
|
huffman@25904
|
42 |
using Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fast
|
huffman@25904
|
43 |
|
huffman@25904
|
44 |
lemma convex_le_PDUnit_PDPlus_iff [simp]:
|
huffman@25904
|
45 |
"(PDUnit a \<le>\<natural> PDPlus t u) = (PDUnit a \<le>\<natural> t \<and> PDUnit a \<le>\<natural> u)"
|
huffman@25904
|
46 |
unfolding convex_le_PDUnit_lemma1 Rep_PDPlus by fast
|
huffman@25904
|
47 |
|
huffman@25904
|
48 |
lemma convex_le_PDUnit_lemma2:
|
huffman@26420
|
49 |
"(t \<le>\<natural> PDUnit b) = (\<forall>a\<in>Rep_pd_basis t. a \<sqsubseteq> b)"
|
huffman@25904
|
50 |
unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit
|
huffman@25904
|
51 |
using Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fast
|
huffman@25904
|
52 |
|
huffman@25904
|
53 |
lemma convex_le_PDPlus_PDUnit_iff [simp]:
|
huffman@25904
|
54 |
"(PDPlus t u \<le>\<natural> PDUnit a) = (t \<le>\<natural> PDUnit a \<and> u \<le>\<natural> PDUnit a)"
|
huffman@25904
|
55 |
unfolding convex_le_PDUnit_lemma2 Rep_PDPlus by fast
|
huffman@25904
|
56 |
|
huffman@25904
|
57 |
lemma convex_le_PDPlus_lemma:
|
huffman@25904
|
58 |
assumes z: "PDPlus t u \<le>\<natural> z"
|
huffman@25904
|
59 |
shows "\<exists>v w. z = PDPlus v w \<and> t \<le>\<natural> v \<and> u \<le>\<natural> w"
|
huffman@25904
|
60 |
proof (intro exI conjI)
|
huffman@26420
|
61 |
let ?A = "{b\<in>Rep_pd_basis z. \<exists>a\<in>Rep_pd_basis t. a \<sqsubseteq> b}"
|
huffman@26420
|
62 |
let ?B = "{b\<in>Rep_pd_basis z. \<exists>a\<in>Rep_pd_basis u. a \<sqsubseteq> b}"
|
huffman@25904
|
63 |
let ?v = "Abs_pd_basis ?A"
|
huffman@25904
|
64 |
let ?w = "Abs_pd_basis ?B"
|
huffman@25904
|
65 |
have Rep_v: "Rep_pd_basis ?v = ?A"
|
huffman@25904
|
66 |
apply (rule Abs_pd_basis_inverse)
|
huffman@25904
|
67 |
apply (rule Rep_pd_basis_nonempty [of t, folded ex_in_conv, THEN exE])
|
huffman@25904
|
68 |
apply (cut_tac z, simp only: convex_le_def lower_le_def, clarify)
|
huffman@25904
|
69 |
apply (drule_tac x=x in bspec, simp add: Rep_PDPlus, erule bexE)
|
huffman@25904
|
70 |
apply (simp add: pd_basis_def)
|
huffman@25904
|
71 |
apply fast
|
huffman@25904
|
72 |
done
|
huffman@25904
|
73 |
have Rep_w: "Rep_pd_basis ?w = ?B"
|
huffman@25904
|
74 |
apply (rule Abs_pd_basis_inverse)
|
huffman@25904
|
75 |
apply (rule Rep_pd_basis_nonempty [of u, folded ex_in_conv, THEN exE])
|
huffman@25904
|
76 |
apply (cut_tac z, simp only: convex_le_def lower_le_def, clarify)
|
huffman@25904
|
77 |
apply (drule_tac x=x in bspec, simp add: Rep_PDPlus, erule bexE)
|
huffman@25904
|
78 |
apply (simp add: pd_basis_def)
|
huffman@25904
|
79 |
apply fast
|
huffman@25904
|
80 |
done
|
huffman@25904
|
81 |
show "z = PDPlus ?v ?w"
|
huffman@25904
|
82 |
apply (insert z)
|
huffman@25904
|
83 |
apply (simp add: convex_le_def, erule conjE)
|
huffman@25904
|
84 |
apply (simp add: Rep_pd_basis_inject [symmetric] Rep_PDPlus)
|
huffman@25904
|
85 |
apply (simp add: Rep_v Rep_w)
|
huffman@25904
|
86 |
apply (rule equalityI)
|
huffman@25904
|
87 |
apply (rule subsetI)
|
huffman@25904
|
88 |
apply (simp only: upper_le_def)
|
huffman@25904
|
89 |
apply (drule (1) bspec, erule bexE)
|
huffman@25904
|
90 |
apply (simp add: Rep_PDPlus)
|
huffman@25904
|
91 |
apply fast
|
huffman@25904
|
92 |
apply fast
|
huffman@25904
|
93 |
done
|
huffman@25904
|
94 |
show "t \<le>\<natural> ?v" "u \<le>\<natural> ?w"
|
huffman@25904
|
95 |
apply (insert z)
|
huffman@25904
|
96 |
apply (simp_all add: convex_le_def upper_le_def lower_le_def Rep_PDPlus Rep_v Rep_w)
|
huffman@25904
|
97 |
apply fast+
|
huffman@25904
|
98 |
done
|
huffman@25904
|
99 |
qed
|
huffman@25904
|
100 |
|
huffman@25904
|
101 |
lemma convex_le_induct [induct set: convex_le]:
|
huffman@25904
|
102 |
assumes le: "t \<le>\<natural> u"
|
huffman@25904
|
103 |
assumes 2: "\<And>t u v. \<lbrakk>P t u; P u v\<rbrakk> \<Longrightarrow> P t v"
|
huffman@26420
|
104 |
assumes 3: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)"
|
huffman@25904
|
105 |
assumes 4: "\<And>t u v w. \<lbrakk>P t v; P u w\<rbrakk> \<Longrightarrow> P (PDPlus t u) (PDPlus v w)"
|
huffman@25904
|
106 |
shows "P t u"
|
huffman@25904
|
107 |
using le apply (induct t arbitrary: u rule: pd_basis_induct)
|
huffman@25904
|
108 |
apply (erule rev_mp)
|
huffman@25904
|
109 |
apply (induct_tac u rule: pd_basis_induct1)
|
huffman@25904
|
110 |
apply (simp add: 3)
|
huffman@25904
|
111 |
apply (simp, clarify, rename_tac a b t)
|
huffman@25904
|
112 |
apply (subgoal_tac "P (PDPlus (PDUnit a) (PDUnit a)) (PDPlus (PDUnit b) t)")
|
huffman@25904
|
113 |
apply (simp add: PDPlus_absorb)
|
huffman@25904
|
114 |
apply (erule (1) 4 [OF 3])
|
huffman@25904
|
115 |
apply (drule convex_le_PDPlus_lemma, clarify)
|
huffman@25904
|
116 |
apply (simp add: 4)
|
huffman@25904
|
117 |
done
|
huffman@25904
|
118 |
|
huffman@27405
|
119 |
lemma pd_take_convex_chain:
|
huffman@27405
|
120 |
"pd_take n t \<le>\<natural> pd_take (Suc n) t"
|
huffman@25904
|
121 |
apply (induct t rule: pd_basis_induct)
|
huffman@27289
|
122 |
apply (simp add: compact_basis.take_chain)
|
huffman@25904
|
123 |
apply (simp add: PDPlus_convex_mono)
|
huffman@25904
|
124 |
done
|
huffman@25904
|
125 |
|
huffman@27405
|
126 |
lemma pd_take_convex_le: "pd_take i t \<le>\<natural> t"
|
huffman@25904
|
127 |
apply (induct t rule: pd_basis_induct)
|
huffman@27289
|
128 |
apply (simp add: compact_basis.take_less)
|
huffman@25904
|
129 |
apply (simp add: PDPlus_convex_mono)
|
huffman@25904
|
130 |
done
|
huffman@25904
|
131 |
|
huffman@27405
|
132 |
lemma pd_take_convex_mono:
|
huffman@27405
|
133 |
"t \<le>\<natural> u \<Longrightarrow> pd_take n t \<le>\<natural> pd_take n u"
|
huffman@25904
|
134 |
apply (erule convex_le_induct)
|
huffman@25904
|
135 |
apply (erule (1) convex_le_trans)
|
huffman@27289
|
136 |
apply (simp add: compact_basis.take_mono)
|
huffman@25904
|
137 |
apply (simp add: PDPlus_convex_mono)
|
huffman@25904
|
138 |
done
|
huffman@25904
|
139 |
|
huffman@25904
|
140 |
|
huffman@25904
|
141 |
subsection {* Type definition *}
|
huffman@25904
|
142 |
|
huffman@27373
|
143 |
typedef (open) 'a convex_pd =
|
huffman@27373
|
144 |
"{S::'a pd_basis set. convex_le.ideal S}"
|
huffman@27373
|
145 |
by (fast intro: convex_le.ideal_principal)
|
huffman@25904
|
146 |
|
huffman@27373
|
147 |
instantiation convex_pd :: (profinite) sq_ord
|
huffman@27373
|
148 |
begin
|
huffman@27373
|
149 |
|
huffman@27373
|
150 |
definition
|
huffman@27373
|
151 |
"x \<sqsubseteq> y \<longleftrightarrow> Rep_convex_pd x \<subseteq> Rep_convex_pd y"
|
huffman@27373
|
152 |
|
huffman@27373
|
153 |
instance ..
|
huffman@27373
|
154 |
end
|
huffman@27373
|
155 |
|
huffman@27373
|
156 |
instance convex_pd :: (profinite) po
|
huffman@27373
|
157 |
by (rule convex_le.typedef_ideal_po
|
huffman@27373
|
158 |
[OF type_definition_convex_pd sq_le_convex_pd_def])
|
huffman@27373
|
159 |
|
huffman@27373
|
160 |
instance convex_pd :: (profinite) cpo
|
huffman@27373
|
161 |
by (rule convex_le.typedef_ideal_cpo
|
huffman@27373
|
162 |
[OF type_definition_convex_pd sq_le_convex_pd_def])
|
huffman@27373
|
163 |
|
huffman@27373
|
164 |
lemma Rep_convex_pd_lub:
|
huffman@27373
|
165 |
"chain Y \<Longrightarrow> Rep_convex_pd (\<Squnion>i. Y i) = (\<Union>i. Rep_convex_pd (Y i))"
|
huffman@27373
|
166 |
by (rule convex_le.typedef_ideal_rep_contlub
|
huffman@27373
|
167 |
[OF type_definition_convex_pd sq_le_convex_pd_def])
|
huffman@27373
|
168 |
|
huffman@27373
|
169 |
lemma ideal_Rep_convex_pd: "convex_le.ideal (Rep_convex_pd xs)"
|
huffman@26927
|
170 |
by (rule Rep_convex_pd [unfolded mem_Collect_eq])
|
huffman@25904
|
171 |
|
huffman@25904
|
172 |
definition
|
huffman@25904
|
173 |
convex_principal :: "'a pd_basis \<Rightarrow> 'a convex_pd" where
|
huffman@27373
|
174 |
"convex_principal t = Abs_convex_pd {u. u \<le>\<natural> t}"
|
huffman@25904
|
175 |
|
huffman@25904
|
176 |
lemma Rep_convex_principal:
|
huffman@27373
|
177 |
"Rep_convex_pd (convex_principal t) = {u. u \<le>\<natural> t}"
|
huffman@25904
|
178 |
unfolding convex_principal_def
|
huffman@27297
|
179 |
by (simp add: Abs_convex_pd_inverse convex_le.ideal_principal)
|
huffman@25904
|
180 |
|
huffman@25904
|
181 |
interpretation convex_pd:
|
huffman@27405
|
182 |
ideal_completion [convex_le pd_take convex_principal Rep_convex_pd]
|
huffman@25904
|
183 |
apply unfold_locales
|
huffman@27405
|
184 |
apply (rule pd_take_convex_le)
|
huffman@27405
|
185 |
apply (rule pd_take_idem)
|
huffman@27405
|
186 |
apply (erule pd_take_convex_mono)
|
huffman@27405
|
187 |
apply (rule pd_take_convex_chain)
|
huffman@27405
|
188 |
apply (rule finite_range_pd_take)
|
huffman@27405
|
189 |
apply (rule pd_take_covers)
|
huffman@26420
|
190 |
apply (rule ideal_Rep_convex_pd)
|
huffman@27373
|
191 |
apply (erule Rep_convex_pd_lub)
|
huffman@26420
|
192 |
apply (rule Rep_convex_principal)
|
huffman@27373
|
193 |
apply (simp only: sq_le_convex_pd_def)
|
huffman@25904
|
194 |
done
|
huffman@25904
|
195 |
|
huffman@27289
|
196 |
text {* Convex powerdomain is pointed *}
|
huffman@25904
|
197 |
|
huffman@25904
|
198 |
lemma convex_pd_minimal: "convex_principal (PDUnit compact_bot) \<sqsubseteq> ys"
|
huffman@25904
|
199 |
by (induct ys rule: convex_pd.principal_induct, simp, simp)
|
huffman@25904
|
200 |
|
huffman@25904
|
201 |
instance convex_pd :: (bifinite) pcpo
|
huffman@26927
|
202 |
by intro_classes (fast intro: convex_pd_minimal)
|
huffman@25904
|
203 |
|
huffman@25904
|
204 |
lemma inst_convex_pd_pcpo: "\<bottom> = convex_principal (PDUnit compact_bot)"
|
huffman@25904
|
205 |
by (rule convex_pd_minimal [THEN UU_I, symmetric])
|
huffman@25904
|
206 |
|
huffman@27289
|
207 |
text {* Convex powerdomain is profinite *}
|
huffman@25904
|
208 |
|
huffman@26962
|
209 |
instantiation convex_pd :: (profinite) profinite
|
huffman@26962
|
210 |
begin
|
huffman@25904
|
211 |
|
huffman@26962
|
212 |
definition
|
huffman@26962
|
213 |
approx_convex_pd_def: "approx = convex_pd.completion_approx"
|
huffman@26927
|
214 |
|
huffman@26962
|
215 |
instance
|
huffman@26927
|
216 |
apply (intro_classes, unfold approx_convex_pd_def)
|
huffman@27310
|
217 |
apply (rule convex_pd.chain_completion_approx)
|
huffman@26927
|
218 |
apply (rule convex_pd.lub_completion_approx)
|
huffman@26927
|
219 |
apply (rule convex_pd.completion_approx_idem)
|
huffman@26927
|
220 |
apply (rule convex_pd.finite_fixes_completion_approx)
|
huffman@26927
|
221 |
done
|
huffman@26927
|
222 |
|
huffman@26962
|
223 |
end
|
huffman@26962
|
224 |
|
huffman@26927
|
225 |
instance convex_pd :: (bifinite) bifinite ..
|
huffman@25904
|
226 |
|
huffman@25904
|
227 |
lemma approx_convex_principal [simp]:
|
huffman@27405
|
228 |
"approx n\<cdot>(convex_principal t) = convex_principal (pd_take n t)"
|
huffman@25904
|
229 |
unfolding approx_convex_pd_def
|
huffman@26927
|
230 |
by (rule convex_pd.completion_approx_principal)
|
huffman@25904
|
231 |
|
huffman@25904
|
232 |
lemma approx_eq_convex_principal:
|
huffman@27405
|
233 |
"\<exists>t\<in>Rep_convex_pd xs. approx n\<cdot>xs = convex_principal (pd_take n t)"
|
huffman@25904
|
234 |
unfolding approx_convex_pd_def
|
huffman@26927
|
235 |
by (rule convex_pd.completion_approx_eq_principal)
|
huffman@26407
|
236 |
|
huffman@25904
|
237 |
|
huffman@26927
|
238 |
subsection {* Monadic unit and plus *}
|
huffman@25904
|
239 |
|
huffman@25904
|
240 |
definition
|
huffman@25904
|
241 |
convex_unit :: "'a \<rightarrow> 'a convex_pd" where
|
huffman@25904
|
242 |
"convex_unit = compact_basis.basis_fun (\<lambda>a. convex_principal (PDUnit a))"
|
huffman@25904
|
243 |
|
huffman@25904
|
244 |
definition
|
huffman@25904
|
245 |
convex_plus :: "'a convex_pd \<rightarrow> 'a convex_pd \<rightarrow> 'a convex_pd" where
|
huffman@25904
|
246 |
"convex_plus = convex_pd.basis_fun (\<lambda>t. convex_pd.basis_fun (\<lambda>u.
|
huffman@25904
|
247 |
convex_principal (PDPlus t u)))"
|
huffman@25904
|
248 |
|
huffman@25904
|
249 |
abbreviation
|
huffman@25904
|
250 |
convex_add :: "'a convex_pd \<Rightarrow> 'a convex_pd \<Rightarrow> 'a convex_pd"
|
huffman@25904
|
251 |
(infixl "+\<natural>" 65) where
|
huffman@25904
|
252 |
"xs +\<natural> ys == convex_plus\<cdot>xs\<cdot>ys"
|
huffman@25904
|
253 |
|
huffman@26927
|
254 |
syntax
|
huffman@26927
|
255 |
"_convex_pd" :: "args \<Rightarrow> 'a convex_pd" ("{_}\<natural>")
|
huffman@26927
|
256 |
|
huffman@26927
|
257 |
translations
|
huffman@26927
|
258 |
"{x,xs}\<natural>" == "{x}\<natural> +\<natural> {xs}\<natural>"
|
huffman@26927
|
259 |
"{x}\<natural>" == "CONST convex_unit\<cdot>x"
|
huffman@26927
|
260 |
|
huffman@26927
|
261 |
lemma convex_unit_Rep_compact_basis [simp]:
|
huffman@26927
|
262 |
"{Rep_compact_basis a}\<natural> = convex_principal (PDUnit a)"
|
huffman@26927
|
263 |
unfolding convex_unit_def
|
huffman@27289
|
264 |
by (simp add: compact_basis.basis_fun_principal PDUnit_convex_mono)
|
huffman@26927
|
265 |
|
huffman@25904
|
266 |
lemma convex_plus_principal [simp]:
|
huffman@26927
|
267 |
"convex_principal t +\<natural> convex_principal u = convex_principal (PDPlus t u)"
|
huffman@25904
|
268 |
unfolding convex_plus_def
|
huffman@25904
|
269 |
by (simp add: convex_pd.basis_fun_principal
|
huffman@25904
|
270 |
convex_pd.basis_fun_mono PDPlus_convex_mono)
|
huffman@25904
|
271 |
|
huffman@26927
|
272 |
lemma approx_convex_unit [simp]:
|
huffman@26927
|
273 |
"approx n\<cdot>{x}\<natural> = {approx n\<cdot>x}\<natural>"
|
huffman@27289
|
274 |
apply (induct x rule: compact_basis.principal_induct, simp)
|
huffman@26927
|
275 |
apply (simp add: approx_Rep_compact_basis)
|
huffman@26927
|
276 |
done
|
huffman@26927
|
277 |
|
huffman@25904
|
278 |
lemma approx_convex_plus [simp]:
|
huffman@26927
|
279 |
"approx n\<cdot>(xs +\<natural> ys) = approx n\<cdot>xs +\<natural> approx n\<cdot>ys"
|
huffman@27289
|
280 |
by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
|
huffman@25904
|
281 |
|
huffman@25904
|
282 |
lemma convex_plus_assoc:
|
huffman@26927
|
283 |
"(xs +\<natural> ys) +\<natural> zs = xs +\<natural> (ys +\<natural> zs)"
|
huffman@27289
|
284 |
apply (induct xs ys arbitrary: zs rule: convex_pd.principal_induct2, simp, simp)
|
huffman@27289
|
285 |
apply (rule_tac x=zs in convex_pd.principal_induct, simp)
|
huffman@25904
|
286 |
apply (simp add: PDPlus_assoc)
|
huffman@25904
|
287 |
done
|
huffman@25904
|
288 |
|
huffman@26927
|
289 |
lemma convex_plus_commute: "xs +\<natural> ys = ys +\<natural> xs"
|
huffman@27289
|
290 |
apply (induct xs ys rule: convex_pd.principal_induct2, simp, simp)
|
huffman@26927
|
291 |
apply (simp add: PDPlus_commute)
|
huffman@26927
|
292 |
done
|
huffman@26927
|
293 |
|
huffman@26927
|
294 |
lemma convex_plus_absorb: "xs +\<natural> xs = xs"
|
huffman@27289
|
295 |
apply (induct xs rule: convex_pd.principal_induct, simp)
|
huffman@25904
|
296 |
apply (simp add: PDPlus_absorb)
|
huffman@25904
|
297 |
done
|
huffman@25904
|
298 |
|
huffman@26927
|
299 |
interpretation aci_convex_plus: ab_semigroup_idem_mult ["op +\<natural>"]
|
huffman@26927
|
300 |
by unfold_locales
|
huffman@26927
|
301 |
(rule convex_plus_assoc convex_plus_commute convex_plus_absorb)+
|
huffman@26927
|
302 |
|
huffman@26927
|
303 |
lemma convex_plus_left_commute: "xs +\<natural> (ys +\<natural> zs) = ys +\<natural> (xs +\<natural> zs)"
|
huffman@26927
|
304 |
by (rule aci_convex_plus.mult_left_commute)
|
huffman@26927
|
305 |
|
huffman@26927
|
306 |
lemma convex_plus_left_absorb: "xs +\<natural> (xs +\<natural> ys) = xs +\<natural> ys"
|
huffman@26927
|
307 |
by (rule aci_convex_plus.mult_left_idem)
|
huffman@26927
|
308 |
|
huffman@26927
|
309 |
lemmas convex_plus_aci = aci_convex_plus.mult_ac_idem
|
huffman@26927
|
310 |
|
huffman@25904
|
311 |
lemma convex_unit_less_plus_iff [simp]:
|
huffman@26927
|
312 |
"{x}\<natural> \<sqsubseteq> ys +\<natural> zs \<longleftrightarrow> {x}\<natural> \<sqsubseteq> ys \<and> {x}\<natural> \<sqsubseteq> zs"
|
huffman@25904
|
313 |
apply (rule iffI)
|
huffman@25904
|
314 |
apply (subgoal_tac
|
huffman@26927
|
315 |
"adm (\<lambda>f. f\<cdot>{x}\<natural> \<sqsubseteq> f\<cdot>ys \<and> f\<cdot>{x}\<natural> \<sqsubseteq> f\<cdot>zs)")
|
huffman@25925
|
316 |
apply (drule admD, rule chain_approx)
|
huffman@25904
|
317 |
apply (drule_tac f="approx i" in monofun_cfun_arg)
|
huffman@27289
|
318 |
apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp)
|
huffman@27289
|
319 |
apply (cut_tac x="approx i\<cdot>ys" in convex_pd.compact_imp_principal, simp)
|
huffman@27289
|
320 |
apply (cut_tac x="approx i\<cdot>zs" in convex_pd.compact_imp_principal, simp)
|
huffman@25904
|
321 |
apply (clarify, simp)
|
huffman@25904
|
322 |
apply simp
|
huffman@25904
|
323 |
apply simp
|
huffman@25904
|
324 |
apply (erule conjE)
|
huffman@26927
|
325 |
apply (subst convex_plus_absorb [of "{x}\<natural>", symmetric])
|
huffman@25904
|
326 |
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
|
huffman@25904
|
327 |
done
|
huffman@25904
|
328 |
|
huffman@25904
|
329 |
lemma convex_plus_less_unit_iff [simp]:
|
huffman@26927
|
330 |
"xs +\<natural> ys \<sqsubseteq> {z}\<natural> \<longleftrightarrow> xs \<sqsubseteq> {z}\<natural> \<and> ys \<sqsubseteq> {z}\<natural>"
|
huffman@25904
|
331 |
apply (rule iffI)
|
huffman@25904
|
332 |
apply (subgoal_tac
|
huffman@26927
|
333 |
"adm (\<lambda>f. f\<cdot>xs \<sqsubseteq> f\<cdot>{z}\<natural> \<and> f\<cdot>ys \<sqsubseteq> f\<cdot>{z}\<natural>)")
|
huffman@25925
|
334 |
apply (drule admD, rule chain_approx)
|
huffman@25904
|
335 |
apply (drule_tac f="approx i" in monofun_cfun_arg)
|
huffman@27289
|
336 |
apply (cut_tac x="approx i\<cdot>xs" in convex_pd.compact_imp_principal, simp)
|
huffman@27289
|
337 |
apply (cut_tac x="approx i\<cdot>ys" in convex_pd.compact_imp_principal, simp)
|
huffman@27289
|
338 |
apply (cut_tac x="approx i\<cdot>z" in compact_basis.compact_imp_principal, simp)
|
huffman@25904
|
339 |
apply (clarify, simp)
|
huffman@25904
|
340 |
apply simp
|
huffman@25904
|
341 |
apply simp
|
huffman@25904
|
342 |
apply (erule conjE)
|
huffman@26927
|
343 |
apply (subst convex_plus_absorb [of "{z}\<natural>", symmetric])
|
huffman@25904
|
344 |
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
|
huffman@25904
|
345 |
done
|
huffman@25904
|
346 |
|
huffman@26927
|
347 |
lemma convex_unit_less_iff [simp]: "{x}\<natural> \<sqsubseteq> {y}\<natural> \<longleftrightarrow> x \<sqsubseteq> y"
|
huffman@26927
|
348 |
apply (rule iffI)
|
huffman@27309
|
349 |
apply (rule profinite_less_ext)
|
huffman@26927
|
350 |
apply (drule_tac f="approx i" in monofun_cfun_arg, simp)
|
huffman@27289
|
351 |
apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp)
|
huffman@27289
|
352 |
apply (cut_tac x="approx i\<cdot>y" in compact_basis.compact_imp_principal, simp)
|
huffman@27289
|
353 |
apply clarsimp
|
huffman@26927
|
354 |
apply (erule monofun_cfun_arg)
|
huffman@26927
|
355 |
done
|
huffman@26927
|
356 |
|
huffman@26927
|
357 |
lemma convex_unit_eq_iff [simp]: "{x}\<natural> = {y}\<natural> \<longleftrightarrow> x = y"
|
huffman@26927
|
358 |
unfolding po_eq_conv by simp
|
huffman@26927
|
359 |
|
huffman@26927
|
360 |
lemma convex_unit_strict [simp]: "{\<bottom>}\<natural> = \<bottom>"
|
huffman@26927
|
361 |
unfolding inst_convex_pd_pcpo Rep_compact_bot [symmetric] by simp
|
huffman@26927
|
362 |
|
huffman@26927
|
363 |
lemma convex_unit_strict_iff [simp]: "{x}\<natural> = \<bottom> \<longleftrightarrow> x = \<bottom>"
|
huffman@26927
|
364 |
unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff)
|
huffman@26927
|
365 |
|
huffman@26927
|
366 |
lemma compact_convex_unit_iff [simp]:
|
huffman@26927
|
367 |
"compact {x}\<natural> \<longleftrightarrow> compact x"
|
huffman@27309
|
368 |
unfolding profinite_compact_iff by simp
|
huffman@26927
|
369 |
|
huffman@26927
|
370 |
lemma compact_convex_plus [simp]:
|
huffman@26927
|
371 |
"\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<natural> ys)"
|
huffman@27289
|
372 |
by (auto dest!: convex_pd.compact_imp_principal)
|
huffman@26927
|
373 |
|
huffman@25904
|
374 |
|
huffman@25904
|
375 |
subsection {* Induction rules *}
|
huffman@25904
|
376 |
|
huffman@25904
|
377 |
lemma convex_pd_induct1:
|
huffman@25904
|
378 |
assumes P: "adm P"
|
huffman@26927
|
379 |
assumes unit: "\<And>x. P {x}\<natural>"
|
huffman@26927
|
380 |
assumes insert: "\<And>x ys. \<lbrakk>P {x}\<natural>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<natural> +\<natural> ys)"
|
huffman@25904
|
381 |
shows "P (xs::'a convex_pd)"
|
huffman@27289
|
382 |
apply (induct xs rule: convex_pd.principal_induct, rule P)
|
huffman@27289
|
383 |
apply (induct_tac a rule: pd_basis_induct1)
|
huffman@25904
|
384 |
apply (simp only: convex_unit_Rep_compact_basis [symmetric])
|
huffman@25904
|
385 |
apply (rule unit)
|
huffman@25904
|
386 |
apply (simp only: convex_unit_Rep_compact_basis [symmetric]
|
huffman@25904
|
387 |
convex_plus_principal [symmetric])
|
huffman@25904
|
388 |
apply (erule insert [OF unit])
|
huffman@25904
|
389 |
done
|
huffman@25904
|
390 |
|
huffman@25904
|
391 |
lemma convex_pd_induct:
|
huffman@25904
|
392 |
assumes P: "adm P"
|
huffman@26927
|
393 |
assumes unit: "\<And>x. P {x}\<natural>"
|
huffman@26927
|
394 |
assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<natural> ys)"
|
huffman@25904
|
395 |
shows "P (xs::'a convex_pd)"
|
huffman@27289
|
396 |
apply (induct xs rule: convex_pd.principal_induct, rule P)
|
huffman@27289
|
397 |
apply (induct_tac a rule: pd_basis_induct)
|
huffman@25904
|
398 |
apply (simp only: convex_unit_Rep_compact_basis [symmetric] unit)
|
huffman@25904
|
399 |
apply (simp only: convex_plus_principal [symmetric] plus)
|
huffman@25904
|
400 |
done
|
huffman@25904
|
401 |
|
huffman@25904
|
402 |
|
huffman@25904
|
403 |
subsection {* Monadic bind *}
|
huffman@25904
|
404 |
|
huffman@25904
|
405 |
definition
|
huffman@25904
|
406 |
convex_bind_basis ::
|
huffman@25904
|
407 |
"'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where
|
huffman@25904
|
408 |
"convex_bind_basis = fold_pd
|
huffman@25904
|
409 |
(\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))
|
huffman@26927
|
410 |
(\<lambda>x y. \<Lambda> f. x\<cdot>f +\<natural> y\<cdot>f)"
|
huffman@25904
|
411 |
|
huffman@26927
|
412 |
lemma ACI_convex_bind:
|
huffman@26927
|
413 |
"ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<natural> y\<cdot>f)"
|
huffman@25904
|
414 |
apply unfold_locales
|
haftmann@26041
|
415 |
apply (simp add: convex_plus_assoc)
|
huffman@25904
|
416 |
apply (simp add: convex_plus_commute)
|
huffman@25904
|
417 |
apply (simp add: convex_plus_absorb eta_cfun)
|
huffman@25904
|
418 |
done
|
huffman@25904
|
419 |
|
huffman@25904
|
420 |
lemma convex_bind_basis_simps [simp]:
|
huffman@25904
|
421 |
"convex_bind_basis (PDUnit a) =
|
huffman@25904
|
422 |
(\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
|
huffman@25904
|
423 |
"convex_bind_basis (PDPlus t u) =
|
huffman@26927
|
424 |
(\<Lambda> f. convex_bind_basis t\<cdot>f +\<natural> convex_bind_basis u\<cdot>f)"
|
huffman@25904
|
425 |
unfolding convex_bind_basis_def
|
huffman@25904
|
426 |
apply -
|
huffman@26927
|
427 |
apply (rule fold_pd_PDUnit [OF ACI_convex_bind])
|
huffman@26927
|
428 |
apply (rule fold_pd_PDPlus [OF ACI_convex_bind])
|
huffman@25904
|
429 |
done
|
huffman@25904
|
430 |
|
huffman@25904
|
431 |
lemma monofun_LAM:
|
huffman@25904
|
432 |
"\<lbrakk>cont f; cont g; \<And>x. f x \<sqsubseteq> g x\<rbrakk> \<Longrightarrow> (\<Lambda> x. f x) \<sqsubseteq> (\<Lambda> x. g x)"
|
huffman@25904
|
433 |
by (simp add: expand_cfun_less)
|
huffman@25904
|
434 |
|
huffman@25904
|
435 |
lemma convex_bind_basis_mono:
|
huffman@25904
|
436 |
"t \<le>\<natural> u \<Longrightarrow> convex_bind_basis t \<sqsubseteq> convex_bind_basis u"
|
huffman@25904
|
437 |
apply (erule convex_le_induct)
|
huffman@25904
|
438 |
apply (erule (1) trans_less)
|
huffman@27289
|
439 |
apply (simp add: monofun_LAM monofun_cfun)
|
huffman@27289
|
440 |
apply (simp add: monofun_LAM monofun_cfun)
|
huffman@25904
|
441 |
done
|
huffman@25904
|
442 |
|
huffman@25904
|
443 |
definition
|
huffman@25904
|
444 |
convex_bind :: "'a convex_pd \<rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where
|
huffman@25904
|
445 |
"convex_bind = convex_pd.basis_fun convex_bind_basis"
|
huffman@25904
|
446 |
|
huffman@25904
|
447 |
lemma convex_bind_principal [simp]:
|
huffman@25904
|
448 |
"convex_bind\<cdot>(convex_principal t) = convex_bind_basis t"
|
huffman@25904
|
449 |
unfolding convex_bind_def
|
huffman@25904
|
450 |
apply (rule convex_pd.basis_fun_principal)
|
huffman@25904
|
451 |
apply (erule convex_bind_basis_mono)
|
huffman@25904
|
452 |
done
|
huffman@25904
|
453 |
|
huffman@25904
|
454 |
lemma convex_bind_unit [simp]:
|
huffman@26927
|
455 |
"convex_bind\<cdot>{x}\<natural>\<cdot>f = f\<cdot>x"
|
huffman@27289
|
456 |
by (induct x rule: compact_basis.principal_induct, simp, simp)
|
huffman@25904
|
457 |
|
huffman@25904
|
458 |
lemma convex_bind_plus [simp]:
|
huffman@26927
|
459 |
"convex_bind\<cdot>(xs +\<natural> ys)\<cdot>f = convex_bind\<cdot>xs\<cdot>f +\<natural> convex_bind\<cdot>ys\<cdot>f"
|
huffman@27289
|
460 |
by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
|
huffman@25904
|
461 |
|
huffman@25904
|
462 |
lemma convex_bind_strict [simp]: "convex_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"
|
huffman@25904
|
463 |
unfolding convex_unit_strict [symmetric] by (rule convex_bind_unit)
|
huffman@25904
|
464 |
|
huffman@25904
|
465 |
|
huffman@25904
|
466 |
subsection {* Map and join *}
|
huffman@25904
|
467 |
|
huffman@25904
|
468 |
definition
|
huffman@25904
|
469 |
convex_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a convex_pd \<rightarrow> 'b convex_pd" where
|
huffman@26927
|
470 |
"convex_map = (\<Lambda> f xs. convex_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<natural>))"
|
huffman@25904
|
471 |
|
huffman@25904
|
472 |
definition
|
huffman@25904
|
473 |
convex_join :: "'a convex_pd convex_pd \<rightarrow> 'a convex_pd" where
|
huffman@25904
|
474 |
"convex_join = (\<Lambda> xss. convex_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"
|
huffman@25904
|
475 |
|
huffman@25904
|
476 |
lemma convex_map_unit [simp]:
|
huffman@25904
|
477 |
"convex_map\<cdot>f\<cdot>(convex_unit\<cdot>x) = convex_unit\<cdot>(f\<cdot>x)"
|
huffman@25904
|
478 |
unfolding convex_map_def by simp
|
huffman@25904
|
479 |
|
huffman@25904
|
480 |
lemma convex_map_plus [simp]:
|
huffman@26927
|
481 |
"convex_map\<cdot>f\<cdot>(xs +\<natural> ys) = convex_map\<cdot>f\<cdot>xs +\<natural> convex_map\<cdot>f\<cdot>ys"
|
huffman@25904
|
482 |
unfolding convex_map_def by simp
|
huffman@25904
|
483 |
|
huffman@25904
|
484 |
lemma convex_join_unit [simp]:
|
huffman@26927
|
485 |
"convex_join\<cdot>{xs}\<natural> = xs"
|
huffman@25904
|
486 |
unfolding convex_join_def by simp
|
huffman@25904
|
487 |
|
huffman@25904
|
488 |
lemma convex_join_plus [simp]:
|
huffman@26927
|
489 |
"convex_join\<cdot>(xss +\<natural> yss) = convex_join\<cdot>xss +\<natural> convex_join\<cdot>yss"
|
huffman@25904
|
490 |
unfolding convex_join_def by simp
|
huffman@25904
|
491 |
|
huffman@25904
|
492 |
lemma convex_map_ident: "convex_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"
|
huffman@25904
|
493 |
by (induct xs rule: convex_pd_induct, simp_all)
|
huffman@25904
|
494 |
|
huffman@25904
|
495 |
lemma convex_map_map:
|
huffman@25904
|
496 |
"convex_map\<cdot>f\<cdot>(convex_map\<cdot>g\<cdot>xs) = convex_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
|
huffman@25904
|
497 |
by (induct xs rule: convex_pd_induct, simp_all)
|
huffman@25904
|
498 |
|
huffman@25904
|
499 |
lemma convex_join_map_unit:
|
huffman@25904
|
500 |
"convex_join\<cdot>(convex_map\<cdot>convex_unit\<cdot>xs) = xs"
|
huffman@25904
|
501 |
by (induct xs rule: convex_pd_induct, simp_all)
|
huffman@25904
|
502 |
|
huffman@25904
|
503 |
lemma convex_join_map_join:
|
huffman@25904
|
504 |
"convex_join\<cdot>(convex_map\<cdot>convex_join\<cdot>xsss) = convex_join\<cdot>(convex_join\<cdot>xsss)"
|
huffman@25904
|
505 |
by (induct xsss rule: convex_pd_induct, simp_all)
|
huffman@25904
|
506 |
|
huffman@25904
|
507 |
lemma convex_join_map_map:
|
huffman@25904
|
508 |
"convex_join\<cdot>(convex_map\<cdot>(convex_map\<cdot>f)\<cdot>xss) =
|
huffman@25904
|
509 |
convex_map\<cdot>f\<cdot>(convex_join\<cdot>xss)"
|
huffman@25904
|
510 |
by (induct xss rule: convex_pd_induct, simp_all)
|
huffman@25904
|
511 |
|
huffman@25904
|
512 |
lemma convex_map_approx: "convex_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs"
|
huffman@25904
|
513 |
by (induct xs rule: convex_pd_induct, simp_all)
|
huffman@25904
|
514 |
|
huffman@25904
|
515 |
|
huffman@25904
|
516 |
subsection {* Conversions to other powerdomains *}
|
huffman@25904
|
517 |
|
huffman@25904
|
518 |
text {* Convex to upper *}
|
huffman@25904
|
519 |
|
huffman@25904
|
520 |
lemma convex_le_imp_upper_le: "t \<le>\<natural> u \<Longrightarrow> t \<le>\<sharp> u"
|
huffman@25904
|
521 |
unfolding convex_le_def by simp
|
huffman@25904
|
522 |
|
huffman@25904
|
523 |
definition
|
huffman@25904
|
524 |
convex_to_upper :: "'a convex_pd \<rightarrow> 'a upper_pd" where
|
huffman@25904
|
525 |
"convex_to_upper = convex_pd.basis_fun upper_principal"
|
huffman@25904
|
526 |
|
huffman@25904
|
527 |
lemma convex_to_upper_principal [simp]:
|
huffman@25904
|
528 |
"convex_to_upper\<cdot>(convex_principal t) = upper_principal t"
|
huffman@25904
|
529 |
unfolding convex_to_upper_def
|
huffman@25904
|
530 |
apply (rule convex_pd.basis_fun_principal)
|
huffman@27289
|
531 |
apply (rule upper_pd.principal_mono)
|
huffman@25904
|
532 |
apply (erule convex_le_imp_upper_le)
|
huffman@25904
|
533 |
done
|
huffman@25904
|
534 |
|
huffman@25904
|
535 |
lemma convex_to_upper_unit [simp]:
|
huffman@26927
|
536 |
"convex_to_upper\<cdot>{x}\<natural> = {x}\<sharp>"
|
huffman@27289
|
537 |
by (induct x rule: compact_basis.principal_induct, simp, simp)
|
huffman@25904
|
538 |
|
huffman@25904
|
539 |
lemma convex_to_upper_plus [simp]:
|
huffman@26927
|
540 |
"convex_to_upper\<cdot>(xs +\<natural> ys) = convex_to_upper\<cdot>xs +\<sharp> convex_to_upper\<cdot>ys"
|
huffman@27289
|
541 |
by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
|
huffman@25904
|
542 |
|
huffman@25904
|
543 |
lemma approx_convex_to_upper:
|
huffman@25904
|
544 |
"approx i\<cdot>(convex_to_upper\<cdot>xs) = convex_to_upper\<cdot>(approx i\<cdot>xs)"
|
huffman@25904
|
545 |
by (induct xs rule: convex_pd_induct, simp, simp, simp)
|
huffman@25904
|
546 |
|
huffman@27289
|
547 |
lemma convex_to_upper_bind [simp]:
|
huffman@27289
|
548 |
"convex_to_upper\<cdot>(convex_bind\<cdot>xs\<cdot>f) =
|
huffman@27289
|
549 |
upper_bind\<cdot>(convex_to_upper\<cdot>xs)\<cdot>(convex_to_upper oo f)"
|
huffman@27289
|
550 |
by (induct xs rule: convex_pd_induct, simp, simp, simp)
|
huffman@27289
|
551 |
|
huffman@27289
|
552 |
lemma convex_to_upper_map [simp]:
|
huffman@27289
|
553 |
"convex_to_upper\<cdot>(convex_map\<cdot>f\<cdot>xs) = upper_map\<cdot>f\<cdot>(convex_to_upper\<cdot>xs)"
|
huffman@27289
|
554 |
by (simp add: convex_map_def upper_map_def cfcomp_LAM)
|
huffman@27289
|
555 |
|
huffman@27289
|
556 |
lemma convex_to_upper_join [simp]:
|
huffman@27289
|
557 |
"convex_to_upper\<cdot>(convex_join\<cdot>xss) =
|
huffman@27289
|
558 |
upper_bind\<cdot>(convex_to_upper\<cdot>xss)\<cdot>convex_to_upper"
|
huffman@27289
|
559 |
by (simp add: convex_join_def upper_join_def cfcomp_LAM eta_cfun)
|
huffman@27289
|
560 |
|
huffman@25904
|
561 |
text {* Convex to lower *}
|
huffman@25904
|
562 |
|
huffman@25904
|
563 |
lemma convex_le_imp_lower_le: "t \<le>\<natural> u \<Longrightarrow> t \<le>\<flat> u"
|
huffman@25904
|
564 |
unfolding convex_le_def by simp
|
huffman@25904
|
565 |
|
huffman@25904
|
566 |
definition
|
huffman@25904
|
567 |
convex_to_lower :: "'a convex_pd \<rightarrow> 'a lower_pd" where
|
huffman@25904
|
568 |
"convex_to_lower = convex_pd.basis_fun lower_principal"
|
huffman@25904
|
569 |
|
huffman@25904
|
570 |
lemma convex_to_lower_principal [simp]:
|
huffman@25904
|
571 |
"convex_to_lower\<cdot>(convex_principal t) = lower_principal t"
|
huffman@25904
|
572 |
unfolding convex_to_lower_def
|
huffman@25904
|
573 |
apply (rule convex_pd.basis_fun_principal)
|
huffman@27289
|
574 |
apply (rule lower_pd.principal_mono)
|
huffman@25904
|
575 |
apply (erule convex_le_imp_lower_le)
|
huffman@25904
|
576 |
done
|
huffman@25904
|
577 |
|
huffman@25904
|
578 |
lemma convex_to_lower_unit [simp]:
|
huffman@26927
|
579 |
"convex_to_lower\<cdot>{x}\<natural> = {x}\<flat>"
|
huffman@27289
|
580 |
by (induct x rule: compact_basis.principal_induct, simp, simp)
|
huffman@25904
|
581 |
|
huffman@25904
|
582 |
lemma convex_to_lower_plus [simp]:
|
huffman@26927
|
583 |
"convex_to_lower\<cdot>(xs +\<natural> ys) = convex_to_lower\<cdot>xs +\<flat> convex_to_lower\<cdot>ys"
|
huffman@27289
|
584 |
by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
|
huffman@25904
|
585 |
|
huffman@25904
|
586 |
lemma approx_convex_to_lower:
|
huffman@25904
|
587 |
"approx i\<cdot>(convex_to_lower\<cdot>xs) = convex_to_lower\<cdot>(approx i\<cdot>xs)"
|
huffman@25904
|
588 |
by (induct xs rule: convex_pd_induct, simp, simp, simp)
|
huffman@25904
|
589 |
|
huffman@27289
|
590 |
lemma convex_to_lower_bind [simp]:
|
huffman@27289
|
591 |
"convex_to_lower\<cdot>(convex_bind\<cdot>xs\<cdot>f) =
|
huffman@27289
|
592 |
lower_bind\<cdot>(convex_to_lower\<cdot>xs)\<cdot>(convex_to_lower oo f)"
|
huffman@27289
|
593 |
by (induct xs rule: convex_pd_induct, simp, simp, simp)
|
huffman@27289
|
594 |
|
huffman@27289
|
595 |
lemma convex_to_lower_map [simp]:
|
huffman@27289
|
596 |
"convex_to_lower\<cdot>(convex_map\<cdot>f\<cdot>xs) = lower_map\<cdot>f\<cdot>(convex_to_lower\<cdot>xs)"
|
huffman@27289
|
597 |
by (simp add: convex_map_def lower_map_def cfcomp_LAM)
|
huffman@27289
|
598 |
|
huffman@27289
|
599 |
lemma convex_to_lower_join [simp]:
|
huffman@27289
|
600 |
"convex_to_lower\<cdot>(convex_join\<cdot>xss) =
|
huffman@27289
|
601 |
lower_bind\<cdot>(convex_to_lower\<cdot>xss)\<cdot>convex_to_lower"
|
huffman@27289
|
602 |
by (simp add: convex_join_def lower_join_def cfcomp_LAM eta_cfun)
|
huffman@27289
|
603 |
|
huffman@25904
|
604 |
text {* Ordering property *}
|
huffman@25904
|
605 |
|
huffman@25904
|
606 |
lemma convex_pd_less_iff:
|
huffman@25904
|
607 |
"(xs \<sqsubseteq> ys) =
|
huffman@25904
|
608 |
(convex_to_upper\<cdot>xs \<sqsubseteq> convex_to_upper\<cdot>ys \<and>
|
huffman@25904
|
609 |
convex_to_lower\<cdot>xs \<sqsubseteq> convex_to_lower\<cdot>ys)"
|
huffman@25904
|
610 |
apply (safe elim!: monofun_cfun_arg)
|
huffman@27309
|
611 |
apply (rule profinite_less_ext)
|
huffman@25904
|
612 |
apply (drule_tac f="approx i" in monofun_cfun_arg)
|
huffman@25904
|
613 |
apply (drule_tac f="approx i" in monofun_cfun_arg)
|
huffman@27289
|
614 |
apply (cut_tac x="approx i\<cdot>xs" in convex_pd.compact_imp_principal, simp)
|
huffman@27289
|
615 |
apply (cut_tac x="approx i\<cdot>ys" in convex_pd.compact_imp_principal, simp)
|
huffman@25904
|
616 |
apply clarify
|
huffman@25904
|
617 |
apply (simp add: approx_convex_to_upper approx_convex_to_lower convex_le_def)
|
huffman@25904
|
618 |
done
|
huffman@25904
|
619 |
|
huffman@26927
|
620 |
lemmas convex_plus_less_plus_iff =
|
huffman@26927
|
621 |
convex_pd_less_iff [where xs="xs +\<natural> ys" and ys="zs +\<natural> ws", standard]
|
huffman@26927
|
622 |
|
huffman@26927
|
623 |
lemmas convex_pd_less_simps =
|
huffman@26927
|
624 |
convex_unit_less_plus_iff
|
huffman@26927
|
625 |
convex_plus_less_unit_iff
|
huffman@26927
|
626 |
convex_plus_less_plus_iff
|
huffman@26927
|
627 |
convex_unit_less_iff
|
huffman@26927
|
628 |
convex_to_upper_unit
|
huffman@26927
|
629 |
convex_to_upper_plus
|
huffman@26927
|
630 |
convex_to_lower_unit
|
huffman@26927
|
631 |
convex_to_lower_plus
|
huffman@26927
|
632 |
upper_pd_less_simps
|
huffman@26927
|
633 |
lower_pd_less_simps
|
huffman@26927
|
634 |
|
huffman@25904
|
635 |
end
|