author | wenzelm |
Sun, 15 Oct 2000 19:50:35 +0200 | |
changeset 10220 | 2a726de6e124 |
parent 9021 | 5643223dad0a |
child 13550 | 5a176b8dda84 |
permissions | -rw-r--r-- |
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(* Title: HOL/UNITY/Constrains |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1998 University of Cambridge |
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|
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Safety relations: restricted to the set of reachable states. |
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*) |
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|
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(*** 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)"; |
|
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by (blast_tac (claset() addIs [stableI, constrainsI] @ reachable.intrs) 1); |
|
25 |
qed "stable_reachable"; |
|
26 |
||
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AddSIs [stable_reachable]; |
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Addsimps [stable_reachable]; |
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|
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); |
|
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by (blast_tac (claset() addIs reachable.intrs) 1); |
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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"; |
|
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||
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||
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(*** Co ***) |
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|
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(*Needed because its operands are sets*) |
49 |
overload_1st_set "Constrains.Constrains"; |
|
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|
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(*F : B co B' ==> F : (reachable F Int B) co (reachable F Int B')*) |
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bind_thm ("constrains_reachable_Int", |
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subset_refl RS |
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rewrite_rule [stable_def] stable_reachable RS |
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constrains_Int); |
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(*Resembles the previous definition of Constrains*) |
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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 |
||
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Goalw [Constrains_def] "F : A co A' ==> F : A Co A'"; |
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by (blast_tac (claset() addIs [constrains_weaken_L]) 1); |
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qed "constrains_imp_Constrains"; |
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|
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Goalw [stable_def, Stable_def] "F : stable A ==> F : Stable A"; |
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by (etac constrains_imp_Constrains 1); |
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qed "stable_imp_Stable"; |
|
71 |
||
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val prems = Goal |
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"(!!act s s'. [| act: Acts F; (s,s') : act; s: A |] ==> s': A') \ |
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\ ==> F : A Co A'"; |
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by (rtac constrains_imp_Constrains 1); |
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by (blast_tac (claset() addIs (constrainsI::prems)) 1); |
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qed "ConstrainsI"; |
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|
6536 | 79 |
Goalw [Constrains_def, constrains_def] "F : {} Co B"; |
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by (Blast_tac 1); |
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qed "Constrains_empty"; |
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|
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Goal "F : A Co UNIV"; |
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by (blast_tac (claset() addIs [ConstrainsI]) 1); |
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qed "Constrains_UNIV"; |
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AddIffs [Constrains_empty, Constrains_UNIV]; |
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|
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Goalw [Constrains_def] |
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"[| F : A Co A'; A'<=B' |] ==> F : A Co B'"; |
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90 |
by (blast_tac (claset() addIs [constrains_weaken_R]) 1); |
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qed "Constrains_weaken_R"; |
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|
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Goalw [Constrains_def] |
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"[| F : A Co A'; B<=A |] ==> F : B Co A'"; |
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by (blast_tac (claset() addIs [constrains_weaken_L]) 1); |
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qed "Constrains_weaken_L"; |
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|
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98 |
Goalw [Constrains_def] |
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"[| F : A Co A'; B<=A; A'<=B' |] ==> F : B Co B'"; |
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by (blast_tac (claset() addIs [constrains_weaken]) 1); |
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qed "Constrains_weaken"; |
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|
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(** Union **) |
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Goalw [Constrains_def] |
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"[| F : A Co A'; F : B Co B' |] \ |
107 |
\ ==> F : (A Un B) Co (A' Un B')"; |
|
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by (blast_tac (claset() addIs [constrains_Un RS constrains_weaken]) 1); |
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qed "Constrains_Un"; |
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val [prem] = Goalw [Constrains_def] |
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112 |
"(!!i. i:I ==> F : (A i) Co (A' i)) \ |
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\ ==> F : (UN i:I. A i) Co (UN i:I. A' i)"; |
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114 |
by (rtac CollectI 1); |
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115 |
by (rtac (prem RS CollectD RS constrains_UN RS constrains_weaken) 1); |
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116 |
by Auto_tac; |
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117 |
qed "Constrains_UN"; |
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|
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(** Intersection **) |
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Goalw [Constrains_def] |
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"[| F : A Co A'; F : B Co B' |] \ |
123 |
\ ==> F : (A Int B) Co (A' Int B')"; |
|
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by (blast_tac (claset() addIs [constrains_Int RS constrains_weaken]) 1); |
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qed "Constrains_Int"; |
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val [prem] = Goalw [Constrains_def] |
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128 |
"(!!i. i:I ==> F : (A i) Co (A' i)) \ |
6536 | 129 |
\ ==> F : (INT i:I. A i) Co (INT i:I. A' i)"; |
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130 |
by (rtac CollectI 1); |
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131 |
by (rtac (prem RS CollectD RS constrains_INT RS constrains_weaken) 1); |
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132 |
by Auto_tac; |
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133 |
qed "Constrains_INT"; |
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134 |
|
6536 | 135 |
Goal "F : A Co A' ==> reachable F Int A <= A'"; |
6575 | 136 |
by (asm_full_simp_tac (simpset() addsimps [constrains_imp_subset, |
137 |
Constrains_def]) 1); |
|
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138 |
qed "Constrains_imp_subset"; |
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139 |
|
6575 | 140 |
Goal "[| F : A Co B; F : B Co C |] ==> F : A Co C"; |
141 |
by (full_simp_tac (simpset() addsimps [Constrains_eq_constrains]) 1); |
|
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142 |
by (blast_tac (claset() addIs [constrains_trans, constrains_weaken]) 1); |
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143 |
qed "Constrains_trans"; |
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144 |
|
6575 | 145 |
Goal "[| F : A Co (A' Un B); F : B Co B' |] ==> F : A Co (A' Un B')"; |
146 |
by (full_simp_tac (simpset() addsimps [Constrains_eq_constrains, |
|
147 |
constrains_def]) 1); |
|
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148 |
by (Blast_tac 1); |
6012
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149 |
qed "Constrains_cancel"; |
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150 |
|
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151 |
|
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152 |
(*** Stable ***) |
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153 |
|
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154 |
(*Useful because there's no Stable_weaken. [Tanja Vos]*) |
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155 |
Goal "[| F: Stable A; A = B |] ==> F : Stable B"; |
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156 |
by (Blast_tac 1); |
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157 |
qed "Stable_eq"; |
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158 |
|
5648 | 159 |
Goal "(F : Stable A) = (F : stable (reachable F Int A))"; |
6575 | 160 |
by (simp_tac (simpset() addsimps [Stable_def, Constrains_eq_constrains, |
161 |
stable_def]) 1); |
|
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162 |
qed "Stable_eq_stable"; |
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163 |
|
6536 | 164 |
Goalw [Stable_def] "F : A Co A ==> F : Stable A"; |
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165 |
by (assume_tac 1); |
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166 |
qed "StableI"; |
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167 |
|
6536 | 168 |
Goalw [Stable_def] "F : Stable A ==> F : A Co A"; |
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169 |
by (assume_tac 1); |
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170 |
qed "StableD"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
171 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
172 |
Goalw [Stable_def] |
5648 | 173 |
"[| 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
|
174 |
by (blast_tac (claset() addIs [Constrains_Un]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
175 |
qed "Stable_Un"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
176 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
177 |
Goalw [Stable_def] |
5648 | 178 |
"[| 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
|
179 |
by (blast_tac (claset() addIs [Constrains_Int]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
180 |
qed "Stable_Int"; |
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 : A Co (C Un A') |] \ |
184 |
\ ==> F : (C Un A) Co (C Un A')"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
185 |
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
|
186 |
qed "Stable_Constrains_Un"; |
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] |
6536 | 189 |
"[| F : Stable C; F : (C Int A) Co A' |] \ |
190 |
\ ==> F : (C Int A) Co (C Int A')"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
191 |
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
|
192 |
qed "Stable_Constrains_Int"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
193 |
|
8334
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
194 |
val [prem] = Goalw [Stable_def] |
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
195 |
"(!!i. i:I ==> F : Stable (A i)) ==> F : Stable (UN i:I. A i)"; |
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
196 |
by (rtac (prem RS Constrains_UN) 1); |
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
197 |
by (assume_tac 1); |
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
198 |
qed "Stable_UN"; |
5804
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
199 |
|
8334
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
200 |
val [prem] = Goalw [Stable_def] |
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
201 |
"(!!i. i:I ==> F : Stable (A i)) ==> F : Stable (INT i:I. A i)"; |
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
202 |
by (rtac (prem RS Constrains_INT) 1); |
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
203 |
by (assume_tac 1); |
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
204 |
qed "Stable_INT"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
205 |
|
5648 | 206 |
Goal "F : Stable (reachable F)"; |
8069
19b9f92ca503
working with weak LeadsTo in guarantees precondition\!
paulson
parents:
7689
diff
changeset
|
207 |
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
|
208 |
qed "Stable_reachable"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
209 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
210 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
211 |
|
5784 | 212 |
(*** Increasing ***) |
213 |
||
8216 | 214 |
Goalw [Increasing_def] |
215 |
"F : Increasing f ==> F : Stable {s. x <= f s}"; |
|
216 |
by (Blast_tac 1); |
|
217 |
qed "IncreasingD"; |
|
218 |
||
5784 | 219 |
Goalw [Increasing_def, Stable_def, Constrains_def, stable_def, constrains_def] |
6704 | 220 |
"mono g ==> Increasing f <= Increasing (g o f)"; |
5784 | 221 |
by Auto_tac; |
6704 | 222 |
by (blast_tac (claset() addIs [monoD, order_trans]) 1); |
223 |
qed "mono_Increasing_o"; |
|
5784 | 224 |
|
225 |
Goalw [Increasing_def] |
|
8216 | 226 |
"!!z::nat. F : Increasing f ==> F: Stable {s. z < f s}"; |
5784 | 227 |
by (simp_tac (simpset() addsimps [Suc_le_eq RS sym]) 1); |
228 |
by (Blast_tac 1); |
|
8216 | 229 |
qed "strict_IncreasingD"; |
5784 | 230 |
|
231 |
Goalw [increasing_def, Increasing_def] |
|
232 |
"F : increasing f ==> F : Increasing f"; |
|
233 |
by (blast_tac (claset() addIs [stable_imp_Stable]) 1); |
|
234 |
qed "increasing_imp_Increasing"; |
|
235 |
||
9021
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
236 |
bind_thm ("Increasing_constant", |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
237 |
increasing_constant RS increasing_imp_Increasing); |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
238 |
AddIffs [Increasing_constant]; |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
239 |
|
5784 | 240 |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
241 |
(*** 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
|
242 |
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
|
243 |
in forward proof. ***) |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
244 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
245 |
Goalw [Constrains_def, constrains_def] |
6536 | 246 |
"[| ALL m. F : {s. s x = m} Co (B m) |] \ |
247 |
\ ==> 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
|
248 |
by (Blast_tac 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
249 |
qed "Elimination"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
250 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
251 |
(*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
|
252 |
state is identified with its one variable.*) |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
253 |
Goalw [Constrains_def, constrains_def] |
6536 | 254 |
"(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
|
255 |
by (Blast_tac 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
256 |
qed "Elimination_sing"; |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
257 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
258 |
|
6570 | 259 |
(*** Specialized laws for handling Always ***) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
260 |
|
6570 | 261 |
(** Natural deduction rules for "Always A" **) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
262 |
|
6570 | 263 |
Goal "[| Init F<=A; F : Stable A |] ==> F : Always A"; |
264 |
by (asm_simp_tac (simpset() addsimps [Always_def]) 1); |
|
265 |
qed "AlwaysI"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
266 |
|
6570 | 267 |
Goal "F : Always A ==> Init F<=A & F : Stable A"; |
268 |
by (asm_full_simp_tac (simpset() addsimps [Always_def]) 1); |
|
269 |
qed "AlwaysD"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
270 |
|
6570 | 271 |
bind_thm ("AlwaysE", AlwaysD RS conjE); |
8334
7896bcbd8641
Added Tanja's Detects and Reachability theories. Also
paulson
parents:
8216
diff
changeset
|
272 |
bind_thm ("Always_imp_Stable", AlwaysD RS conjunct2); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
273 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
274 |
|
6570 | 275 |
(*The set of all reachable states is Always*) |
276 |
Goal "F : Always A ==> reachable F <= A"; |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
277 |
by (full_simp_tac |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
278 |
(simpset() addsimps [Stable_def, Constrains_def, constrains_def, |
6570 | 279 |
Always_def]) 1); |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
280 |
by (rtac subsetI 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
281 |
by (etac reachable.induct 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
282 |
by (REPEAT (blast_tac (claset() addIs reachable.intrs) 1)); |
6570 | 283 |
qed "Always_includes_reachable"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
284 |
|
6575 | 285 |
Goalw [Always_def, invariant_def, Stable_def, stable_def] |
6570 | 286 |
"F : invariant A ==> F : Always A"; |
6575 | 287 |
by (blast_tac (claset() addIs [constrains_imp_Constrains]) 1); |
6570 | 288 |
qed "invariant_imp_Always"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
289 |
|
6672 | 290 |
bind_thm ("Always_reachable", invariant_reachable RS invariant_imp_Always); |
291 |
||
6575 | 292 |
Goal "Always A = {F. F : invariant (reachable F Int A)}"; |
293 |
by (simp_tac (simpset() addsimps [Always_def, invariant_def, Stable_def, |
|
294 |
Constrains_eq_constrains, stable_def]) 1); |
|
5648 | 295 |
by (blast_tac (claset() addIs reachable.intrs) 1); |
6570 | 296 |
qed "Always_eq_invariant_reachable"; |
5648 | 297 |
|
6570 | 298 |
(*the RHS is the traditional definition of the "always" operator*) |
299 |
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
|
300 |
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
|
301 |
simpset() addsimps [Int_absorb2, invariant_reachable, |
6570 | 302 |
Always_eq_invariant_reachable])); |
303 |
qed "Always_eq_includes_reachable"; |
|
5648 | 304 |
|
7689 | 305 |
Goal "Always UNIV = UNIV"; |
306 |
by (auto_tac (claset(), |
|
307 |
simpset() addsimps [Always_eq_includes_reachable])); |
|
308 |
qed "Always_UNIV_eq"; |
|
309 |
Addsimps [Always_UNIV_eq]; |
|
5648 | 310 |
|
9021
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
311 |
Goal "UNIV <= A ==> F : Always A"; |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
312 |
by (auto_tac (claset(), simpset() addsimps [Always_eq_includes_reachable])); |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
313 |
qed "UNIV_AlwaysI"; |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
314 |
|
6570 | 315 |
Goal "Always A = (UN I: Pow A. invariant I)"; |
316 |
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
|
317 |
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
|
318 |
impOfSubs invariant_includes_reachable]) 1); |
6570 | 319 |
qed "Always_eq_UN_invariant"; |
320 |
||
321 |
Goal "[| F : Always A; A <= B |] ==> F : Always B"; |
|
322 |
by (auto_tac (claset(), simpset() addsimps [Always_eq_includes_reachable])); |
|
323 |
qed "Always_weaken"; |
|
5804
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
324 |
|
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
5784
diff
changeset
|
325 |
|
6570 | 326 |
(*** "Co" rules involving Always ***) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
327 |
|
6739
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
328 |
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
|
329 |
by (asm_simp_tac |
6570 | 330 |
(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
|
331 |
Constrains_def, Int_assoc RS sym]) 1); |
6739
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
332 |
qed "Always_Constrains_pre"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
333 |
|
6739
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
334 |
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
|
335 |
by (asm_simp_tac |
6570 | 336 |
(simpset() addsimps [Always_includes_reachable RS Int_absorb2, |
6575 | 337 |
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
|
338 |
qed "Always_Constrains_post"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
339 |
|
6739
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
340 |
(* [| 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
|
341 |
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
|
342 |
|
66e4118eead9
replaced rules Always_ConstrainsI/D by equivalences Always_Constrains_pre,
paulson
parents:
6704
diff
changeset
|
343 |
(* [| 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
|
344 |
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
|
345 |
|
9021
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
346 |
(*The analogous proof of Always_LeadsTo_weaken doesn't terminate*) |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
347 |
Goal "[| F : Always C; F : A Co A'; \ |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
348 |
\ C Int B <= A; C Int A' <= B' |] \ |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
349 |
\ ==> F : B Co B'"; |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
350 |
by (rtac Always_ConstrainsI 1); |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
351 |
by (assume_tac 1); |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
352 |
by (dtac Always_ConstrainsD 1); |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
353 |
by (assume_tac 1); |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
354 |
by (blast_tac (claset() addIs [Constrains_weaken]) 1); |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
355 |
qed "Always_Constrains_weaken"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
356 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
357 |
|
6570 | 358 |
(** Conjoining Always properties **) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
359 |
|
7541
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
360 |
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
|
361 |
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
|
362 |
qed "Always_Int_distrib"; |
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
363 |
|
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
364 |
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
|
365 |
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
|
366 |
qed "Always_INT_distrib"; |
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
367 |
|
6570 | 368 |
Goal "[| F : Always A; F : Always B |] ==> F : Always (A Int B)"; |
9021
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
369 |
by (asm_simp_tac (simpset() addsimps [Always_Int_distrib]) 1); |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
370 |
qed "Always_Int_I"; |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
371 |
|
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
372 |
(*Allows a kind of "implication introduction"*) |
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
373 |
Goal "F : Always A ==> (F : Always (-A Un B)) = (F : Always B)"; |
6570 | 374 |
by (auto_tac (claset(), simpset() addsimps [Always_eq_includes_reachable])); |
9021
5643223dad0a
new theorems Always_Constrains_weaken and Always_Compl_Un_eq
paulson
parents:
8334
diff
changeset
|
375 |
qed "Always_Compl_Un_eq"; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
376 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
377 |
(*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
|
378 |
used by Always_Int_I) *) |
6570 | 379 |
val Always_thin = |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
380 |
read_instantiate_sg (sign_of thy) |
6570 | 381 |
[("V", "?F : Always ?A")] thin_rl; |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
382 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
383 |
(*Combines two invariance ASSUMPTIONS into one. USEFUL??*) |
7541
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
384 |
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
|
385 |
|
5319
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5313
diff
changeset
|
386 |
(*Combines a list of invariance THEOREMS into one.*) |
7541
1a7a38d8f5bd
new theorem Always_INT_distrib; therefore renamed Always_Int
paulson
parents:
7403
diff
changeset
|
387 |
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
|
388 |
|
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
diff
changeset
|
389 |
|
5648 | 390 |
(*To allow expansion of the program's definition when appropriate*) |
391 |
val program_defs_ref = ref ([] : thm list); |
|
392 |
||
6536 | 393 |
(*proves "co" properties when the program is specified*) |
5648 | 394 |
fun constrains_tac i = |
5422 | 395 |
SELECT_GOAL |
7403 | 396 |
(EVERY [REPEAT (Always_Int_tac 1), |
397 |
REPEAT (etac Always_ConstrainsI 1 |
|
398 |
ORELSE |
|
399 |
resolve_tac [StableI, stableI, |
|
5422 | 400 |
constrains_imp_Constrains] 1), |
401 |
rtac constrainsI 1, |
|
7403 | 402 |
full_simp_tac (simpset() addsimps !program_defs_ref) 1, |
5620 | 403 |
REPEAT (FIRSTGOAL (etac disjE)), |
5422 | 404 |
ALLGOALS Clarify_tac, |
5648 | 405 |
ALLGOALS Asm_full_simp_tac]) i; |
7403 | 406 |
|
407 |
||
408 |
(*For proving invariants*) |
|
409 |
fun always_tac i = |
|
410 |
rtac AlwaysI i THEN Force_tac i THEN constrains_tac i; |