author | paulson |
Tue, 21 Dec 1999 15:03:02 +0100 | |
changeset 8069 | 19b9f92ca503 |
parent 7689 | affe0c2fdfbf |
child 8216 | e4b3192dfefa |
permissions | -rw-r--r-- |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
1 |
(* Title: HOL/UNITY/Constrains |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
2 |
ID: $Id$ |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
4 |
Copyright 1998 University of Cambridge |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
5 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
6 |
Safety relations: restricted to the set of reachable states. |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
7 |
*) |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
8 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
9 |
|
6535 | 10 |
(*** traces and reachable ***) |
11 |
||
12 |
Goal "reachable F = {s. EX evs. (s,evs): traces (Init F) (Acts F)}"; |
|
13 |
by Safe_tac; |
|
14 |
by (etac traces.induct 2); |
|
15 |
by (etac reachable.induct 1); |
|
16 |
by (ALLGOALS (blast_tac (claset() addIs reachable.intrs @ traces.intrs))); |
|
17 |
qed "reachable_equiv_traces"; |
|
18 |
||
19 |
Goal "Init F <= reachable F"; |
|
20 |
by (blast_tac (claset() addIs reachable.intrs) 1); |
|
21 |
qed "Init_subset_reachable"; |
|
22 |
||
23 |
Goal "Acts G <= Acts F ==> G : stable (reachable F)"; |
|
24 |
by (blast_tac (claset() addIs [stableI, constrainsI] @ reachable.intrs) 1); |
|
25 |
qed "stable_reachable"; |
|
26 |
||
8069
19b9f92ca503
working with weak LeadsTo in guarantees precondition\!
paulson
parents:
7689
diff
changeset
|
27 |
AddSIs [stable_reachable]; |
19b9f92ca503
working with weak LeadsTo in guarantees precondition\!
paulson
parents:
7689
diff
changeset
|
28 |
Addsimps [stable_reachable]; |
6535 | 29 |
|
30 |
(*The set of all reachable states is an invariant...*) |
|
31 |
Goal "F : invariant (reachable F)"; |
|
32 |
by (simp_tac (simpset() addsimps [invariant_def]) 1); |
|
8069
19b9f92ca503
working with weak LeadsTo in guarantees precondition\!
paulson
parents:
7689
diff
changeset
|
33 |
by (blast_tac (claset() addIs reachable.intrs) 1); |
6535 | 34 |
qed "invariant_reachable"; |
35 |
||
36 |
(*...in fact the strongest invariant!*) |
|
37 |
Goal "F : invariant A ==> reachable F <= A"; |
|
38 |
by (full_simp_tac |
|
39 |
(simpset() addsimps [stable_def, constrains_def, invariant_def]) 1); |
|
40 |
by (rtac subsetI 1); |
|
41 |
by (etac reachable.induct 1); |
|
42 |
by (REPEAT (blast_tac (claset() addIs reachable.intrs) 1)); |
|
43 |
qed "invariant_includes_reachable"; |
|
44 |
||
45 |
||
6536 | 46 |
(*** Co ***) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
47 |
|
7403 | 48 |
(*Needed because its operands are sets*) |
49 |
overload_1st_set "Constrains.Constrains"; |
|
5340 | 50 |
|
6536 | 51 |
(*F : B co B' ==> F : (reachable F Int B) co (reachable F Int B')*) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
52 |
bind_thm ("constrains_reachable_Int", |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
53 |
subset_refl RS |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
54 |
rewrite_rule [stable_def] stable_reachable RS |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
55 |
constrains_Int); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
56 |
|
6575 | 57 |
(*Resembles the previous definition of Constrains*) |
58 |
Goalw [Constrains_def] |
|
59 |
"A Co B = {F. F : (reachable F Int A) co (reachable F Int B)}"; |
|
60 |
by (blast_tac (claset() addDs [constrains_reachable_Int] |
|
61 |
addIs [constrains_weaken]) 1); |
|
62 |
qed "Constrains_eq_constrains"; |
|
63 |
||
6536 | 64 |
Goalw [Constrains_def] "F : A co A' ==> F : A Co A'"; |
6575 | 65 |
by (blast_tac (claset() addIs [constrains_weaken_L]) 1); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
66 |
qed "constrains_imp_Constrains"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
67 |
|
5648 | 68 |
Goalw [stable_def, Stable_def] "F : stable A ==> F : Stable A"; |
5631 | 69 |
by (etac constrains_imp_Constrains 1); |
70 |
qed "stable_imp_Stable"; |
|
71 |
||
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
72 |
val prems = Goal |
5620 | 73 |
"(!!act s s'. [| act: Acts F; (s,s') : act; s: A |] ==> s': A') \ |
6536 | 74 |
\ ==> F : A Co A'"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
75 |
by (rtac constrains_imp_Constrains 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
76 |
by (blast_tac (claset() addIs (constrainsI::prems)) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
77 |
qed "ConstrainsI"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
78 |
|
6536 | 79 |
Goalw [Constrains_def, constrains_def] "F : {} Co B"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
80 |
by (Blast_tac 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
81 |
qed "Constrains_empty"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
82 |
|
6536 | 83 |
Goal "F : A Co UNIV"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
84 |
by (blast_tac (claset() addIs [ConstrainsI]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
85 |
qed "Constrains_UNIV"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
86 |
AddIffs [Constrains_empty, Constrains_UNIV]; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
87 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
88 |
Goalw [Constrains_def] |
6536 | 89 |
"[| F : A Co A'; A'<=B' |] ==> F : A Co B'"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
90 |
by (blast_tac (claset() addIs [constrains_weaken_R]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
91 |
qed "Constrains_weaken_R"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
92 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
93 |
Goalw [Constrains_def] |
6536 | 94 |
"[| F : A Co A'; B<=A |] ==> F : B Co A'"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
95 |
by (blast_tac (claset() addIs [constrains_weaken_L]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
96 |
qed "Constrains_weaken_L"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
97 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
98 |
Goalw [Constrains_def] |
6536 | 99 |
"[| F : A Co A'; B<=A; A'<=B' |] ==> F : B Co B'"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
100 |
by (blast_tac (claset() addIs [constrains_weaken]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
101 |
qed "Constrains_weaken"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
102 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
103 |
(** Union **) |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
104 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
105 |
Goalw [Constrains_def] |
6536 | 106 |
"[| F : A Co A'; F : B Co B' |] \ |
107 |
\ ==> F : (A Un B) Co (A' Un B')"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
108 |
by (blast_tac (claset() addIs [constrains_Un RS constrains_weaken]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
109 |
qed "Constrains_Un"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
110 |
|
6536 | 111 |
Goal "ALL i:I. F : (A i) Co (A' i) \ |
112 |
\ ==> F : (UN i:I. A i) Co (UN i:I. A' i)"; |
|
5648 | 113 |
by (asm_full_simp_tac (simpset() addsimps [Constrains_def]) 1); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
114 |
by (dtac ball_constrains_UN 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
115 |
by (blast_tac (claset() addIs [constrains_weaken]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
116 |
qed "ball_Constrains_UN"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
117 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
118 |
(** Intersection **) |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
119 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
120 |
Goalw [Constrains_def] |
6536 | 121 |
"[| F : A Co A'; F : B Co B' |] \ |
122 |
\ ==> F : (A Int B) Co (A' Int B')"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
123 |
by (blast_tac (claset() addIs [constrains_Int RS constrains_weaken]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
124 |
qed "Constrains_Int"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
125 |
|
6536 | 126 |
Goal "ALL i:I. F : (A i) Co (A' i) \ |
127 |
\ ==> F : (INT i:I. A i) Co (INT i:I. A' i)"; |
|
5648 | 128 |
by (asm_full_simp_tac (simpset() addsimps [Constrains_def]) 1); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
129 |
by (dtac ball_constrains_INT 1); |
5340 | 130 |
by (dtac constrains_reachable_Int 1); |
131 |
by (blast_tac (claset() addIs [constrains_weaken]) 1); |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
132 |
qed "ball_Constrains_INT"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
133 |
|
6536 | 134 |
Goal "F : A Co A' ==> reachable F Int A <= A'"; |
6575 | 135 |
by (asm_full_simp_tac (simpset() addsimps [constrains_imp_subset, |
136 |
Constrains_def]) 1); |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
137 |
qed "Constrains_imp_subset"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
138 |
|
6575 | 139 |
Goal "[| F : A Co B; F : B Co C |] ==> F : A Co C"; |
140 |
by (full_simp_tac (simpset() addsimps [Constrains_eq_constrains]) 1); |
|
6295
351b3c2b0d83
removed the infernal States, eqStates, compatible, etc.
paulson
parents:
6012
diff
changeset
|
141 |
by (blast_tac (claset() addIs [constrains_trans, constrains_weaken]) 1); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
142 |
qed "Constrains_trans"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
143 |
|
6575 | 144 |
Goal "[| F : A Co (A' Un B); F : B Co B' |] ==> F : A Co (A' Un B')"; |
145 |
by (full_simp_tac (simpset() addsimps [Constrains_eq_constrains, |
|
146 |
constrains_def]) 1); |
|
6295
351b3c2b0d83
removed the infernal States, eqStates, compatible, etc.
paulson
parents:
6012
diff
changeset
|
147 |
by (Blast_tac 1); |
6012
1894bfc4aee9
Addition of the States component; parts of Comp not working
paulson
parents:
5804
diff
changeset
|
148 |
qed "Constrains_cancel"; |
1894bfc4aee9
Addition of the States component; parts of Comp not working
paulson
parents:
5804
diff
changeset
|
149 |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
150 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
151 |
(*** Stable ***) |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
152 |
|
5648 | 153 |
Goal "(F : Stable A) = (F : stable (reachable F Int A))"; |
6575 | 154 |
by (simp_tac (simpset() addsimps [Stable_def, Constrains_eq_constrains, |
155 |
stable_def]) 1); |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
156 |
qed "Stable_eq_stable"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
157 |
|
6536 | 158 |
Goalw [Stable_def] "F : A Co A ==> F : Stable A"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
159 |
by (assume_tac 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
160 |
qed "StableI"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
161 |
|
6536 | 162 |
Goalw [Stable_def] "F : Stable A ==> F : A Co A"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
163 |
by (assume_tac 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
164 |
qed "StableD"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
165 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
166 |
Goalw [Stable_def] |
5648 | 167 |
"[| F : Stable A; F : Stable A' |] ==> F : Stable (A Un A')"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
168 |
by (blast_tac (claset() addIs [Constrains_Un]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
169 |
qed "Stable_Un"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
170 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
171 |
Goalw [Stable_def] |
5648 | 172 |
"[| F : Stable A; F : Stable A' |] ==> F : Stable (A Int A')"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
173 |
by (blast_tac (claset() addIs [Constrains_Int]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
174 |
qed "Stable_Int"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
175 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
176 |
Goalw [Stable_def] |
6536 | 177 |
"[| F : Stable C; F : A Co (C Un A') |] \ |
178 |
\ ==> F : (C Un A) Co (C Un A')"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
179 |
by (blast_tac (claset() addIs [Constrains_Un RS Constrains_weaken]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
180 |
qed "Stable_Constrains_Un"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
181 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
182 |
Goalw [Stable_def] |
6536 | 183 |
"[| F : Stable C; F : (C Int A) Co A' |] \ |
184 |
\ ==> F : (C Int A) Co (C Int A')"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
185 |
by (blast_tac (claset() addIs [Constrains_Int RS Constrains_weaken]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
186 |
qed "Stable_Constrains_Int"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
187 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
188 |
Goalw [Stable_def] |
5804
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
189 |
"(ALL i:I. F : Stable (A i)) ==> F : Stable (UN i:I. A i)"; |
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
190 |
by (etac ball_Constrains_UN 1); |
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
191 |
qed "ball_Stable_UN"; |
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
192 |
|
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
193 |
Goalw [Stable_def] |
5648 | 194 |
"(ALL i:I. F : Stable (A i)) ==> F : Stable (INT i:I. A i)"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
195 |
by (etac ball_Constrains_INT 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
196 |
qed "ball_Stable_INT"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
197 |
|
5648 | 198 |
Goal "F : Stable (reachable F)"; |
8069
19b9f92ca503
working with weak LeadsTo in guarantees precondition\!
paulson
parents:
7689
diff
changeset
|
199 |
by (simp_tac (simpset() addsimps [Stable_eq_stable]) 1); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
200 |
qed "Stable_reachable"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
201 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
202 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
203 |
|
5784 | 204 |
(*** Increasing ***) |
205 |
||
206 |
Goalw [Increasing_def, Stable_def, Constrains_def, stable_def, constrains_def] |
|
6704 | 207 |
"mono g ==> Increasing f <= Increasing (g o f)"; |
5784 | 208 |
by Auto_tac; |
6704 | 209 |
by (blast_tac (claset() addIs [monoD, order_trans]) 1); |
210 |
qed "mono_Increasing_o"; |
|
5784 | 211 |
|
212 |
Goalw [Increasing_def] |
|
213 |
"Increasing f <= {F. ALL z::nat. F: Stable {s. z < f s}}"; |
|
214 |
by (simp_tac (simpset() addsimps [Suc_le_eq RS sym]) 1); |
|
215 |
by (Blast_tac 1); |
|
5804
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
216 |
qed "Increasing_Stable_less"; |
5784 | 217 |
|
218 |
Goalw [increasing_def, Increasing_def] |
|
219 |
"F : increasing f ==> F : Increasing f"; |
|
220 |
by (blast_tac (claset() addIs [stable_imp_Stable]) 1); |
|
221 |
qed "increasing_imp_Increasing"; |
|
222 |
||
223 |
||
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
224 |
(*** The Elimination Theorem. The "free" m has become universally quantified! |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
225 |
Should the premise be !!m instead of ALL m ? Would make it harder to use |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
226 |
in forward proof. ***) |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
227 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
228 |
Goalw [Constrains_def, constrains_def] |
6536 | 229 |
"[| ALL m. F : {s. s x = m} Co (B m) |] \ |
230 |
\ ==> F : {s. s x : M} Co (UN m:M. B m)"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
231 |
by (Blast_tac 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
232 |
qed "Elimination"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
233 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
234 |
(*As above, but for the trivial case of a one-variable state, in which the |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
235 |
state is identified with its one variable.*) |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
236 |
Goalw [Constrains_def, constrains_def] |
6536 | 237 |
"(ALL m. F : {m} Co (B m)) ==> F : M Co (UN m:M. B m)"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
238 |
by (Blast_tac 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
239 |
qed "Elimination_sing"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
240 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
241 |
|
6570 | 242 |
(*** Specialized laws for handling Always ***) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
243 |
|
6570 | 244 |
(** Natural deduction rules for "Always A" **) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
245 |
|
6570 | 246 |
Goal "[| Init F<=A; F : Stable A |] ==> F : Always A"; |
247 |
by (asm_simp_tac (simpset() addsimps [Always_def]) 1); |
|
248 |
qed "AlwaysI"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
249 |
|
6570 | 250 |
Goal "F : Always A ==> Init F<=A & F : Stable A"; |
251 |
by (asm_full_simp_tac (simpset() addsimps [Always_def]) 1); |
|
252 |
qed "AlwaysD"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
253 |
|
6570 | 254 |
bind_thm ("AlwaysE", AlwaysD RS conjE); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
255 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
256 |
|
6570 | 257 |
(*The set of all reachable states is Always*) |
258 |
Goal "F : Always A ==> reachable F <= A"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
259 |
by (full_simp_tac |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
260 |
(simpset() addsimps [Stable_def, Constrains_def, constrains_def, |
6570 | 261 |
Always_def]) 1); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
262 |
by (rtac subsetI 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
263 |
by (etac reachable.induct 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
264 |
by (REPEAT (blast_tac (claset() addIs reachable.intrs) 1)); |
6570 | 265 |
qed "Always_includes_reachable"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
266 |
|
6575 | 267 |
Goalw [Always_def, invariant_def, Stable_def, stable_def] |
6570 | 268 |
"F : invariant A ==> F : Always A"; |
6575 | 269 |
by (blast_tac (claset() addIs [constrains_imp_Constrains]) 1); |
6570 | 270 |
qed "invariant_imp_Always"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
271 |
|
6672 | 272 |
bind_thm ("Always_reachable", invariant_reachable RS invariant_imp_Always); |
273 |
||
6575 | 274 |
Goal "Always A = {F. F : invariant (reachable F Int A)}"; |
275 |
by (simp_tac (simpset() addsimps [Always_def, invariant_def, Stable_def, |
|
276 |
Constrains_eq_constrains, stable_def]) 1); |
|
5648 | 277 |
by (blast_tac (claset() addIs reachable.intrs) 1); |
6570 | 278 |
qed "Always_eq_invariant_reachable"; |
5648 | 279 |
|
6570 | 280 |
(*the RHS is the traditional definition of the "always" operator*) |
281 |
Goal "Always A = {F. reachable F <= A}"; |
|
5804
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
282 |
by (auto_tac (claset() addDs [invariant_includes_reachable], |
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
283 |
simpset() addsimps [Int_absorb2, invariant_reachable, |
6570 | 284 |
Always_eq_invariant_reachable])); |
285 |
qed "Always_eq_includes_reachable"; |
|
5648 | 286 |
|
7689 | 287 |
Goal "Always UNIV = UNIV"; |
288 |
by (auto_tac (claset(), |
|
289 |
simpset() addsimps [Always_eq_includes_reachable])); |
|
290 |
qed "Always_UNIV_eq"; |
|
291 |
Addsimps [Always_UNIV_eq]; |
|
5648 | 292 |
|
6570 | 293 |
Goal "Always A = (UN I: Pow A. invariant I)"; |
294 |
by (simp_tac (simpset() addsimps [Always_eq_includes_reachable]) 1); |
|
5804
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
295 |
by (blast_tac (claset() addIs [invariantI, impOfSubs Init_subset_reachable, |
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
296 |
impOfSubs invariant_includes_reachable]) 1); |
6570 | 297 |
qed "Always_eq_UN_invariant"; |
298 |
||
299 |
Goal "[| F : Always A; A <= B |] ==> F : Always B"; |
|
300 |
by (auto_tac (claset(), simpset() addsimps [Always_eq_includes_reachable])); |
|
301 |
qed "Always_weaken"; |
|
5804
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
302 |
|
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
303 |
|
6570 | 304 |
(*** "Co" rules involving Always ***) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
305 |
|
6739
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
306 |
Goal "F : Always INV ==> (F : (INV Int A) Co A') = (F : A Co A')"; |
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
307 |
by (asm_simp_tac |
6570 | 308 |
(simpset() addsimps [Always_includes_reachable RS Int_absorb2, |
5804
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
309 |
Constrains_def, Int_assoc RS sym]) 1); |
6739
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
310 |
qed "Always_Constrains_pre"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
311 |
|
6739
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
312 |
Goal "F : Always INV ==> (F : A Co (INV Int A')) = (F : A Co A')"; |
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
313 |
by (asm_simp_tac |
6570 | 314 |
(simpset() addsimps [Always_includes_reachable RS Int_absorb2, |
6575 | 315 |
Constrains_eq_constrains, Int_assoc RS sym]) 1); |
6739
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
316 |
qed "Always_Constrains_post"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
317 |
|
6739
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
318 |
(* [| F : Always INV; F : (INV Int A) Co A' |] ==> F : A Co A' *) |
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
319 |
bind_thm ("Always_ConstrainsI", Always_Constrains_pre RS iffD1); |
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
320 |
|
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
321 |
(* [| F : Always INV; F : A Co A' |] ==> F : A Co (INV Int A') *) |
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
322 |
bind_thm ("Always_ConstrainsD", Always_Constrains_post RS iffD2); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
323 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
324 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
325 |
|
6570 | 326 |
(** Conjoining Always properties **) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
327 |
|
7541
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
328 |
Goal "Always (A Int B) = Always A Int Always B"; |
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
329 |
by (auto_tac (claset(), simpset() addsimps [Always_eq_includes_reachable])); |
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
330 |
qed "Always_Int_distrib"; |
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
331 |
|
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
332 |
Goal "Always (INTER I A) = (INT i:I. Always (A i))"; |
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
333 |
by (auto_tac (claset(), simpset() addsimps [Always_eq_includes_reachable])); |
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
334 |
qed "Always_INT_distrib"; |
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
335 |
|
6570 | 336 |
Goal "[| F : Always A; F : Always B |] ==> F : Always (A Int B)"; |
337 |
by (auto_tac (claset(), simpset() addsimps [Always_eq_includes_reachable])); |
|
7541
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
338 |
qed "Always_Int_I"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
339 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
340 |
(*Delete the nearest invariance assumption (which will be the second one |
7541
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
341 |
used by Always_Int_I) *) |
6570 | 342 |
val Always_thin = |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
343 |
read_instantiate_sg (sign_of thy) |
6570 | 344 |
[("V", "?F : Always ?A")] thin_rl; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
345 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
346 |
(*Combines two invariance ASSUMPTIONS into one. USEFUL??*) |
7541
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
347 |
val Always_Int_tac = dtac Always_Int_I THEN' assume_tac THEN' etac Always_thin; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
348 |
|
5319
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5313
diff
changeset
|
349 |
(*Combines a list of invariance THEOREMS into one.*) |
7541
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
350 |
val Always_Int_rule = foldr1 (fn (th1,th2) => [th1,th2] MRS Always_Int_I); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
351 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
352 |
|
5648 | 353 |
(*To allow expansion of the program's definition when appropriate*) |
354 |
val program_defs_ref = ref ([] : thm list); |
|
355 |
||
6536 | 356 |
(*proves "co" properties when the program is specified*) |
5648 | 357 |
fun constrains_tac i = |
5422 | 358 |
SELECT_GOAL |
7403 | 359 |
(EVERY [REPEAT (Always_Int_tac 1), |
360 |
REPEAT (etac Always_ConstrainsI 1 |
|
361 |
ORELSE |
|
362 |
resolve_tac [StableI, stableI, |
|
5422 | 363 |
constrains_imp_Constrains] 1), |
364 |
rtac constrainsI 1, |
|
7403 | 365 |
full_simp_tac (simpset() addsimps !program_defs_ref) 1, |
5620 | 366 |
REPEAT (FIRSTGOAL (etac disjE)), |
5422 | 367 |
ALLGOALS Clarify_tac, |
5648 | 368 |
ALLGOALS Asm_full_simp_tac]) i; |
7403 | 369 |
|
370 |
||
371 |
(*For proving invariants*) |
|
372 |
fun always_tac i = |
|
373 |
rtac AlwaysI i THEN Force_tac i THEN constrains_tac i; |