6706
|
1 |
(* Title: HOL/UNITY/Follows
|
|
2 |
ID: $Id$
|
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
|
|
4 |
Copyright 1998 University of Cambridge
|
|
5 |
|
8128
|
6 |
The "Follows" relation of Charpentier and Sivilotte
|
6706
|
7 |
*)
|
|
8 |
|
13796
|
9 |
theory Follows = SubstAx + ListOrder + Multiset:
|
6706
|
10 |
|
|
11 |
constdefs
|
|
12 |
|
|
13 |
Follows :: "['a => 'b::{order}, 'a => 'b::{order}] => 'a program set"
|
6809
|
14 |
(infixl "Fols" 65)
|
|
15 |
"f Fols g == Increasing g Int Increasing f Int
|
|
16 |
Always {s. f s <= g s} Int
|
|
17 |
(INT k. {s. k <= g s} LeadsTo {s. k <= f s})"
|
6706
|
18 |
|
|
19 |
|
13796
|
20 |
(*Does this hold for "invariant"?*)
|
|
21 |
lemma mono_Always_o:
|
|
22 |
"mono h ==> Always {s. f s <= g s} <= Always {s. h (f s) <= h (g s)}"
|
|
23 |
apply (simp add: Always_eq_includes_reachable)
|
|
24 |
apply (blast intro: monoD)
|
|
25 |
done
|
|
26 |
|
|
27 |
lemma mono_LeadsTo_o:
|
|
28 |
"mono (h::'a::order => 'b::order)
|
|
29 |
==> (INT j. {s. j <= g s} LeadsTo {s. j <= f s}) <=
|
|
30 |
(INT k. {s. k <= h (g s)} LeadsTo {s. k <= h (f s)})"
|
|
31 |
apply auto
|
|
32 |
apply (rule single_LeadsTo_I)
|
|
33 |
apply (drule_tac x = "g s" in spec)
|
|
34 |
apply (erule LeadsTo_weaken)
|
|
35 |
apply (blast intro: monoD order_trans)+
|
|
36 |
done
|
|
37 |
|
|
38 |
lemma Follows_constant: "F : (%s. c) Fols (%s. c)"
|
|
39 |
by (unfold Follows_def, auto)
|
|
40 |
declare Follows_constant [iff]
|
|
41 |
|
|
42 |
lemma mono_Follows_o: "mono h ==> f Fols g <= (h o f) Fols (h o g)"
|
|
43 |
apply (unfold Follows_def, clarify)
|
|
44 |
apply (simp add: mono_Increasing_o [THEN [2] rev_subsetD]
|
|
45 |
mono_Always_o [THEN [2] rev_subsetD]
|
|
46 |
mono_LeadsTo_o [THEN [2] rev_subsetD, THEN INT_D])
|
|
47 |
done
|
|
48 |
|
|
49 |
lemma mono_Follows_apply:
|
|
50 |
"mono h ==> f Fols g <= (%x. h (f x)) Fols (%x. h (g x))"
|
|
51 |
apply (drule mono_Follows_o)
|
|
52 |
apply (force simp add: o_def)
|
|
53 |
done
|
|
54 |
|
|
55 |
lemma Follows_trans:
|
|
56 |
"[| F : f Fols g; F: g Fols h |] ==> F : f Fols h"
|
|
57 |
apply (unfold Follows_def)
|
|
58 |
apply (simp add: Always_eq_includes_reachable)
|
|
59 |
apply (blast intro: order_trans LeadsTo_Trans)
|
|
60 |
done
|
|
61 |
|
|
62 |
|
|
63 |
(** Destructiom rules **)
|
|
64 |
|
|
65 |
lemma Follows_Increasing1:
|
|
66 |
"F : f Fols g ==> F : Increasing f"
|
|
67 |
|
|
68 |
apply (unfold Follows_def, blast)
|
|
69 |
done
|
|
70 |
|
|
71 |
lemma Follows_Increasing2:
|
|
72 |
"F : f Fols g ==> F : Increasing g"
|
|
73 |
apply (unfold Follows_def, blast)
|
|
74 |
done
|
|
75 |
|
|
76 |
lemma Follows_Bounded:
|
|
77 |
"F : f Fols g ==> F : Always {s. f s <= g s}"
|
|
78 |
apply (unfold Follows_def, blast)
|
|
79 |
done
|
|
80 |
|
|
81 |
lemma Follows_LeadsTo:
|
|
82 |
"F : f Fols g ==> F : {s. k <= g s} LeadsTo {s. k <= f s}"
|
|
83 |
apply (unfold Follows_def, blast)
|
|
84 |
done
|
|
85 |
|
|
86 |
lemma Follows_LeadsTo_pfixLe:
|
|
87 |
"F : f Fols g ==> F : {s. k pfixLe g s} LeadsTo {s. k pfixLe f s}"
|
|
88 |
apply (rule single_LeadsTo_I, clarify)
|
|
89 |
apply (drule_tac k="g s" in Follows_LeadsTo)
|
|
90 |
apply (erule LeadsTo_weaken)
|
|
91 |
apply blast
|
|
92 |
apply (blast intro: pfixLe_trans prefix_imp_pfixLe)
|
|
93 |
done
|
|
94 |
|
|
95 |
lemma Follows_LeadsTo_pfixGe:
|
|
96 |
"F : f Fols g ==> F : {s. k pfixGe g s} LeadsTo {s. k pfixGe f s}"
|
|
97 |
apply (rule single_LeadsTo_I, clarify)
|
|
98 |
apply (drule_tac k="g s" in Follows_LeadsTo)
|
|
99 |
apply (erule LeadsTo_weaken)
|
|
100 |
apply blast
|
|
101 |
apply (blast intro: pfixGe_trans prefix_imp_pfixGe)
|
|
102 |
done
|
|
103 |
|
|
104 |
|
|
105 |
lemma Always_Follows1:
|
|
106 |
"[| F : Always {s. f s = f' s}; F : f Fols g |] ==> F : f' Fols g"
|
|
107 |
|
|
108 |
apply (unfold Follows_def Increasing_def Stable_def, auto)
|
|
109 |
apply (erule_tac [3] Always_LeadsTo_weaken)
|
|
110 |
apply (erule_tac A = "{s. z <= f s}" and A' = "{s. z <= f s}" in Always_Constrains_weaken, auto)
|
|
111 |
apply (drule Always_Int_I, assumption)
|
|
112 |
apply (force intro: Always_weaken)
|
|
113 |
done
|
|
114 |
|
|
115 |
lemma Always_Follows2:
|
|
116 |
"[| F : Always {s. g s = g' s}; F : f Fols g |] ==> F : f Fols g'"
|
|
117 |
apply (unfold Follows_def Increasing_def Stable_def, auto)
|
|
118 |
apply (erule_tac [3] Always_LeadsTo_weaken)
|
|
119 |
apply (erule_tac A = "{s. z <= g s}" and A' = "{s. z <= g s}" in Always_Constrains_weaken, auto)
|
|
120 |
apply (drule Always_Int_I, assumption)
|
|
121 |
apply (force intro: Always_weaken)
|
|
122 |
done
|
|
123 |
|
|
124 |
|
|
125 |
(** Union properties (with the subset ordering) **)
|
|
126 |
|
|
127 |
(*Can replace "Un" by any sup. But existing max only works for linorders.*)
|
|
128 |
lemma increasing_Un:
|
|
129 |
"[| F : increasing f; F: increasing g |]
|
|
130 |
==> F : increasing (%s. (f s) Un (g s))"
|
|
131 |
apply (unfold increasing_def stable_def constrains_def, auto)
|
|
132 |
apply (drule_tac x = "f xa" in spec)
|
|
133 |
apply (drule_tac x = "g xa" in spec)
|
|
134 |
apply (blast dest!: bspec)
|
|
135 |
done
|
|
136 |
|
|
137 |
lemma Increasing_Un:
|
|
138 |
"[| F : Increasing f; F: Increasing g |]
|
|
139 |
==> F : Increasing (%s. (f s) Un (g s))"
|
|
140 |
apply (unfold Increasing_def Stable_def Constrains_def stable_def constrains_def, auto)
|
|
141 |
apply (drule_tac x = "f xa" in spec)
|
|
142 |
apply (drule_tac x = "g xa" in spec)
|
|
143 |
apply (blast dest!: bspec)
|
|
144 |
done
|
|
145 |
|
|
146 |
|
|
147 |
lemma Always_Un:
|
|
148 |
"[| F : Always {s. f' s <= f s}; F : Always {s. g' s <= g s} |]
|
|
149 |
==> F : Always {s. f' s Un g' s <= f s Un g s}"
|
|
150 |
apply (simp add: Always_eq_includes_reachable, blast)
|
|
151 |
done
|
|
152 |
|
|
153 |
(*Lemma to re-use the argument that one variable increases (progress)
|
|
154 |
while the other variable doesn't decrease (safety)*)
|
|
155 |
lemma Follows_Un_lemma:
|
|
156 |
"[| F : Increasing f; F : Increasing g;
|
|
157 |
F : Increasing g'; F : Always {s. f' s <= f s};
|
|
158 |
ALL k. F : {s. k <= f s} LeadsTo {s. k <= f' s} |]
|
|
159 |
==> F : {s. k <= f s Un g s} LeadsTo {s. k <= f' s Un g s}"
|
|
160 |
apply (rule single_LeadsTo_I)
|
|
161 |
apply (drule_tac x = "f s" in IncreasingD)
|
|
162 |
apply (drule_tac x = "g s" in IncreasingD)
|
|
163 |
apply (rule LeadsTo_weaken)
|
|
164 |
apply (rule PSP_Stable)
|
|
165 |
apply (erule_tac x = "f s" in spec)
|
|
166 |
apply (erule Stable_Int, assumption)
|
|
167 |
apply blast
|
|
168 |
apply blast
|
|
169 |
done
|
|
170 |
|
|
171 |
lemma Follows_Un:
|
|
172 |
"[| F : f' Fols f; F: g' Fols g |]
|
|
173 |
==> F : (%s. (f' s) Un (g' s)) Fols (%s. (f s) Un (g s))"
|
|
174 |
apply (unfold Follows_def)
|
|
175 |
apply (simp add: Increasing_Un Always_Un, auto)
|
|
176 |
apply (rule LeadsTo_Trans)
|
|
177 |
apply (blast intro: Follows_Un_lemma)
|
|
178 |
(*Weakening is used to exchange Un's arguments*)
|
|
179 |
apply (blast intro: Follows_Un_lemma [THEN LeadsTo_weaken])
|
|
180 |
done
|
|
181 |
|
|
182 |
|
|
183 |
(** Multiset union properties (with the multiset ordering) **)
|
|
184 |
|
|
185 |
lemma increasing_union:
|
|
186 |
"[| F : increasing f; F: increasing g |]
|
|
187 |
==> F : increasing (%s. (f s) + (g s :: ('a::order) multiset))"
|
|
188 |
|
|
189 |
apply (unfold increasing_def stable_def constrains_def, auto)
|
|
190 |
apply (drule_tac x = "f xa" in spec)
|
|
191 |
apply (drule_tac x = "g xa" in spec)
|
|
192 |
apply (drule bspec, assumption)
|
|
193 |
apply (blast intro: union_le_mono order_trans)
|
|
194 |
done
|
|
195 |
|
|
196 |
lemma Increasing_union:
|
|
197 |
"[| F : Increasing f; F: Increasing g |]
|
|
198 |
==> F : Increasing (%s. (f s) + (g s :: ('a::order) multiset))"
|
|
199 |
apply (unfold Increasing_def Stable_def Constrains_def stable_def constrains_def, auto)
|
|
200 |
apply (drule_tac x = "f xa" in spec)
|
|
201 |
apply (drule_tac x = "g xa" in spec)
|
|
202 |
apply (drule bspec, assumption)
|
|
203 |
apply (blast intro: union_le_mono order_trans)
|
|
204 |
done
|
|
205 |
|
|
206 |
lemma Always_union:
|
|
207 |
"[| F : Always {s. f' s <= f s}; F : Always {s. g' s <= g s} |]
|
|
208 |
==> F : Always {s. f' s + g' s <= f s + (g s :: ('a::order) multiset)}"
|
|
209 |
apply (simp add: Always_eq_includes_reachable)
|
|
210 |
apply (blast intro: union_le_mono)
|
|
211 |
done
|
|
212 |
|
|
213 |
(*Except the last line, IDENTICAL to the proof script for Follows_Un_lemma*)
|
|
214 |
lemma Follows_union_lemma:
|
|
215 |
"[| F : Increasing f; F : Increasing g;
|
|
216 |
F : Increasing g'; F : Always {s. f' s <= f s};
|
|
217 |
ALL k::('a::order) multiset.
|
|
218 |
F : {s. k <= f s} LeadsTo {s. k <= f' s} |]
|
|
219 |
==> F : {s. k <= f s + g s} LeadsTo {s. k <= f' s + g s}"
|
|
220 |
apply (rule single_LeadsTo_I)
|
|
221 |
apply (drule_tac x = "f s" in IncreasingD)
|
|
222 |
apply (drule_tac x = "g s" in IncreasingD)
|
|
223 |
apply (rule LeadsTo_weaken)
|
|
224 |
apply (rule PSP_Stable)
|
|
225 |
apply (erule_tac x = "f s" in spec)
|
|
226 |
apply (erule Stable_Int, assumption)
|
|
227 |
apply blast
|
|
228 |
apply (blast intro: union_le_mono order_trans)
|
|
229 |
done
|
|
230 |
|
|
231 |
(*The !! is there to influence to effect of permutative rewriting at the end*)
|
|
232 |
lemma Follows_union:
|
|
233 |
"!!g g' ::'b => ('a::order) multiset.
|
|
234 |
[| F : f' Fols f; F: g' Fols g |]
|
|
235 |
==> F : (%s. (f' s) + (g' s)) Fols (%s. (f s) + (g s))"
|
|
236 |
apply (unfold Follows_def)
|
|
237 |
apply (simp add: Increasing_union Always_union, auto)
|
|
238 |
apply (rule LeadsTo_Trans)
|
|
239 |
apply (blast intro: Follows_union_lemma)
|
|
240 |
(*now exchange union's arguments*)
|
|
241 |
apply (simp add: union_commute)
|
|
242 |
apply (blast intro: Follows_union_lemma)
|
|
243 |
done
|
|
244 |
|
|
245 |
lemma Follows_setsum:
|
|
246 |
"!!f ::['c,'b] => ('a::order) multiset.
|
|
247 |
[| ALL i: I. F : f' i Fols f i; finite I |]
|
|
248 |
==> F : (%s. \<Sum>i:I. f' i s) Fols (%s. \<Sum>i:I. f i s)"
|
|
249 |
apply (erule rev_mp)
|
|
250 |
apply (erule finite_induct, simp)
|
|
251 |
apply (simp add: Follows_union)
|
|
252 |
done
|
|
253 |
|
|
254 |
|
|
255 |
(*Currently UNUSED, but possibly of interest*)
|
|
256 |
lemma Increasing_imp_Stable_pfixGe:
|
|
257 |
"F : Increasing func ==> F : Stable {s. h pfixGe (func s)}"
|
|
258 |
apply (simp add: Increasing_def Stable_def Constrains_def constrains_def)
|
|
259 |
apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD]
|
|
260 |
prefix_imp_pfixGe)
|
|
261 |
done
|
|
262 |
|
|
263 |
(*Currently UNUSED, but possibly of interest*)
|
|
264 |
lemma LeadsTo_le_imp_pfixGe:
|
|
265 |
"ALL z. F : {s. z <= f s} LeadsTo {s. z <= g s}
|
|
266 |
==> F : {s. z pfixGe f s} LeadsTo {s. z pfixGe g s}"
|
|
267 |
apply (rule single_LeadsTo_I)
|
|
268 |
apply (drule_tac x = "f s" in spec)
|
|
269 |
apply (erule LeadsTo_weaken)
|
|
270 |
prefer 2
|
|
271 |
apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD]
|
|
272 |
prefix_imp_pfixGe, blast)
|
|
273 |
done
|
|
274 |
|
6706
|
275 |
end
|