--- a/src/HOL/UNITY/Union.ML Wed Dec 02 16:14:09 1998 +0100
+++ b/src/HOL/UNITY/Union.ML Thu Dec 03 10:45:06 1998 +0100
@@ -8,55 +8,124 @@
From Misra's Chapter 5: Asynchronous Compositions of Programs
*)
+Goal "k:I ==> (INT i:I. A i) Int A k = (INT i:I. A i)";
+by (Blast_tac 1);
+qed "INT_absorb2";
+
+
+val rinst = read_instantiate_sg (sign_of thy);
+
+Addcongs [UN_cong, INT_cong];
(** SKIP **)
-Goal "Init SKIP = UNIV";
+Goal "States (SKIP A) = A";
+by (simp_tac (simpset() addsimps [SKIP_def]) 1);
+qed "States_SKIP";
+
+Goal "Init (SKIP A) = A";
by (simp_tac (simpset() addsimps [SKIP_def]) 1);
qed "Init_SKIP";
-Goal "Acts SKIP = {Id}";
+Goal "Acts (SKIP A) = {diag A}";
by (simp_tac (simpset() addsimps [SKIP_def]) 1);
qed "Acts_SKIP";
-Addsimps [Init_SKIP, Acts_SKIP];
+Addsimps [States_SKIP, Init_SKIP, Acts_SKIP];
-Goal "reachable SKIP = UNIV";
-by (force_tac (claset() addIs reachable.intrs, simpset()) 1);
+Goal "reachable (SKIP A) = A";
+by (force_tac (claset() addEs [reachable.induct]
+ addIs reachable.intrs, simpset()) 1);
qed "reachable_SKIP";
Addsimps [reachable_SKIP];
-(** Join and JN **)
+(** Join **)
+
+Goal "States (F Join G) = States F Int States G";
+by (simp_tac (simpset() addsimps [Join_def]) 1);
+qed "States_Join";
Goal "Init (F Join G) = Init F Int Init G";
-by (simp_tac (simpset() addsimps [Join_def]) 1);
+by (auto_tac (claset() addIs [impOfSubs Init_subset_States],
+ simpset() addsimps [Join_def]));
qed "Init_Join";
-Goal "Acts (F Join G) = Acts F Un Acts G";
+Goal "States F = States G ==> Acts (F Join G) = Acts F Un Acts G";
+by (subgoal_tac "Acts F Un Acts G <= Pow (States G Times States G)" 1);
+by (blast_tac (claset() addEs [equalityE]
+ addDs [impOfSubs Acts_subset_Pow_States]) 2);
by (auto_tac (claset(), simpset() addsimps [Join_def]));
qed "Acts_Join";
+
+(** JN **)
+
+Goalw [JOIN_def, SKIP_def] "(JN i:{}. F i) = (SKIP UNIV)";
+by Auto_tac;
+qed "JN_empty";
+Addsimps [JN_empty];
+
+Goal "States (JN i:I. F i) = (INT i:I. States (F i))";
+by (simp_tac (simpset() addsimps [JOIN_def]) 1);
+qed "States_JN";
+
Goal "Init (JN i:I. F i) = (INT i:I. Init (F i))";
-by (simp_tac (simpset() addsimps [JOIN_def]) 1);
+by (auto_tac (claset() addIs [impOfSubs Init_subset_States],
+ simpset() addsimps [JOIN_def]));
qed "Init_JN";
-Goal "Acts (JN i:I. F i) = insert Id (UN i:I. Acts (F i))";
+Goalw [eqStates_def]
+ "[| i: I; eqStates I F |] ==> Acts (JN i:I. F i) = (UN i:I. Acts (F i))";
+by (Clarify_tac 1);
+by (subgoal_tac "(UN i:I. Acts (F i)) <= Pow (St Times St)" 1);
+by (blast_tac (claset() addEs [equalityE]
+ addDs [impOfSubs Acts_subset_Pow_States]) 2);
by (auto_tac (claset(), simpset() addsimps [JOIN_def]));
qed "Acts_JN";
-Addsimps [Init_Join, Init_JN];
+Goal "eqStates I F \
+\ ==> Acts (JN i:I. F i) = \
+\ (if I={} then {diag UNIV} else (UN i:I. Acts (F i)))";
+by (force_tac (claset(), simpset() addsimps [Acts_JN]) 1);
+qed "Acts_JN_if";
+
+Addsimps [States_Join, Init_Join, States_JN, Init_JN];
+
-Goalw [JOIN_def, SKIP_def] "(JN i:{}. F i) = SKIP";
-by Auto_tac;
-qed "JN_empty";
+Goal "(JN x:insert j I. F x) = (F j) Join (JN x:I. F x)";
+by (rtac program_equalityI 1);
+by (ALLGOALS Asm_simp_tac);
+by (asm_simp_tac
+ (simpset() addsimps [JOIN_def, Join_def, Acts_eq_raw,
+ image_UNION, image_Un, image_image, Int_assoc]) 1);
+qed "JN_insert";
+Addsimps[JN_insert];
-Goal "(JN x:insert a A. B x) = (B a) Join (JN x:A. B x)";
-by (rtac program_equalityI 1);
-by (ALLGOALS (simp_tac (simpset() addsimps [JOIN_def, Join_def])));
-qed "JN_insert";
-Addsimps[JN_empty, JN_insert];
+Goal "k:I ==> A k Join (JN i:I. A i) = (JN i:I. A i)";
+by (stac (JN_insert RS sym) 1);
+by (asm_simp_tac (simpset() addsimps [insert_absorb]) 1);
+qed "JN_absorb";
+
+
+Goalw [eqStates_def] "eqStates {} F";
+by (Simp_tac 1);
+qed "eqStates_empty";
+
+Goalw [eqStates_def]
+ "[| eqStates I F; States (F i) = (INT i: I. States (F i)) |] \
+\ ==> eqStates (insert i I) F";
+by Auto_tac;
+qed "eqStates_insertI";
+
+Goalw [eqStates_def]
+ "eqStates (insert i I) F = \
+\ (eqStates I F & (I={} | States (F i) = (INT i: I. States (F i))))";
+by Auto_tac;
+qed "eqStates_insert_eq";
+
+Addsimps [eqStates_empty, eqStates_insert_eq];
(** Algebraic laws **)
@@ -66,170 +135,224 @@
qed "Join_commute";
Goal "(F Join G) Join H = F Join (G Join H)";
-by (simp_tac (simpset() addsimps Un_ac@[Join_def, Int_assoc]) 1);
+by (rtac program_equalityI 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps Un_ac@[Int_assoc])));
+by (asm_simp_tac
+ (simpset() addsimps Un_ac@Int_ac@[Join_def, Acts_eq_raw,
+ image_Un, image_image]) 1);
+by (Blast_tac 1);
qed "Join_assoc";
-Goalw [Join_def, SKIP_def] "SKIP Join F = F";
+Goalw [Join_def, SKIP_def] "States F <= A ==> (SKIP A) Join F = F";
by (rtac program_equalityI 1);
-by (ALLGOALS (simp_tac (simpset() addsimps [insert_absorb])));
+by (ALLGOALS
+ (asm_simp_tac (simpset() addsimps
+ Int_ac@[Acts_subset_Pow_States RS restrict_rel_image,
+ Acts_eq_raw, insert_absorb, Int_absorb1])));
+by (blast_tac (claset() addIs [impOfSubs Init_subset_States]) 1);
qed "Join_SKIP_left";
-Goalw [Join_def, SKIP_def] "F Join SKIP = F";
-by (rtac program_equalityI 1);
-by (ALLGOALS (simp_tac (simpset() addsimps [insert_absorb])));
+Goal "States F <= A ==> F Join (SKIP A) = F";
+by (stac Join_commute 1);
+by (asm_simp_tac (simpset() addsimps [Join_SKIP_left]) 1);
qed "Join_SKIP_right";
Addsimps [Join_SKIP_left, Join_SKIP_right];
Goalw [Join_def] "F Join F = F";
by (rtac program_equalityI 1);
-by Auto_tac;
+by (ALLGOALS
+ (asm_simp_tac (simpset() addsimps
+ [insert_absorb, Acts_subset_Pow_States RS Acts_eq])));
+by (blast_tac (claset() addIs [impOfSubs Init_subset_States]) 1);
qed "Join_absorb";
Addsimps [Join_absorb];
-
-Goal "i : I ==> (JN i:I. A i Join B) = ((JN i:I. A i) Join B)";
-by (auto_tac (claset() addSIs [program_equalityI],
- simpset() addsimps [Acts_JN, Acts_Join]));
-qed "JN_Join_miniscope";
-
-Goal "k:I ==> A k Join (JN i:I. A i) = (JN i:I. A i)";
-by (auto_tac (claset() addSIs [program_equalityI],
- simpset() addsimps [Acts_JN, Acts_Join]));
-qed "JN_absorb";
-
Goal "(JN i: I Un J. A i) = ((JN i: I. A i) Join (JN i:J. A i))";
-by (auto_tac (claset() addSIs [program_equalityI],
- simpset() addsimps [Acts_JN, Acts_Join]));
+by (rtac program_equalityI 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps [INT_Un])));
+by (asm_simp_tac
+ (simpset() addsimps Int_ac@
+ [JOIN_def, Join_def, Acts_eq_raw, UN_Un, INT_Un,
+ image_UNION, image_Un, image_image]) 1);
qed "JN_Join";
Goal "i: I ==> (JN i:I. c) = c";
by (auto_tac (claset() addSIs [program_equalityI],
- simpset() addsimps [Acts_JN]));
+ simpset() addsimps [Acts_JN, eqStates_def]));
qed "JN_constant";
Goal "(JN i:I. A i Join B i) = (JN i:I. A i) Join (JN i:I. B i)";
-by (auto_tac (claset() addSIs [program_equalityI],
- simpset() addsimps [Acts_JN, Acts_Join]));
+by (rtac program_equalityI 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps [INT_Int_distrib])));
+by (asm_simp_tac
+ (simpset() addsimps [JOIN_def, Join_def, Acts_eq_raw, image_UNION,
+ rinst [("A","%x. States (A x) Int States (B x)")]
+ INT_absorb2,
+ image_Un, image_image]) 1);
+by (asm_simp_tac (simpset() addsimps [INT_Int_distrib RS sym] @ Int_ac) 1);
+by (Blast_tac 1);
qed "JN_Join_distrib";
+Goal "i : I ==> (JN i:I. A i Join B) = ((JN i:I. A i) Join B)";
+by (asm_simp_tac (simpset() addsimps [JN_Join_distrib, JN_constant]) 1);
+qed "JN_Join_miniscope";
+
(*** Safety: constrains, stable, FP ***)
-Goalw [constrains_def, JOIN_def]
- "i : I ==> \
-\ (JN i:I. F i) : constrains A B = (ALL i:I. F i : constrains A B)";
-by (Simp_tac 1);
-by (Blast_tac 1);
-qed "constrains_JN";
+Goal "[| F : constrains A B; G : constrains A B |] \
+\ ==> F Join G : constrains A B";
+by (auto_tac (claset(),
+ simpset() addsimps [constrains_def, Join_def, Acts_eq_raw,
+ image_Un]));
+by (REPEAT (force_tac (claset() addSEs [ballE], simpset()) 1));
+qed "constrains_imp_Join_constrains";
-Goal "F Join G : constrains A B = \
-\ (F : constrains A B & G : constrains A B)";
-by (auto_tac
- (claset(),
- simpset() addsimps [constrains_def, Join_def]));
+Goal "States F = States G ==> \
+\ F Join G : constrains A B = (F : constrains A B & G : constrains A B)";
+by (auto_tac (claset(), simpset() addsimps [constrains_def, Acts_Join]));
qed "constrains_Join";
-(**FAILS, I think; see 5.4.1, Substitution Axiom.
- The problem is to relate reachable (F Join G) with
- reachable F and reachable G
+Goal "[| i : I; ALL i:I. F i : constrains A B |] \
+\ ==> (JN i:I. F i) : constrains A B";
+by (full_simp_tac (simpset() addsimps [constrains_def, JOIN_def, Acts_eq_raw,
+ image_Un]) 1);
+by Safe_tac;
+by (REPEAT (force_tac (claset() addSEs [ballE], simpset()) 1));
+qed "constrains_imp_JN_constrains";
-Goalw [Constrains_def]
- "(JN i:I. F i) : Constrains A B = (ALL i:I. F i : Constrains A B)";
-by (simp_tac (simpset() addsimps [constrains_JN]) 1);
-by (Blast_tac 1);
-qed "Constrains_JN";
+Goal "[| i : I; eqStates I F |] \
+\ ==> (JN i:I. F i) : constrains A B = (ALL i:I. F i : constrains A B)";
+by (auto_tac (claset(), simpset() addsimps [constrains_def, Acts_JN]));
+qed "constrains_JN";
+
+ (**FAILS, I think; see 5.4.1, Substitution Axiom.
+ The problem is to relate reachable (F Join G) with
+ reachable F and reachable G
-Goal "F Join G : Constrains A B = (Constrains F A B & Constrains G A B)";
-by (auto_tac
- (claset(),
- simpset() addsimps [Constrains_def, constrains_Join]));
-qed "Constrains_Join";
-**)
+ Goalw [Constrains_def]
+ "(JN i:I. F i) : Constrains A B = (ALL i:I. F i : Constrains A B)";
+ by (simp_tac (simpset() addsimps [constrains_JN]) 1);
+ by (Blast_tac 1);
+ qed "Constrains_JN";
-Goal "[| F : constrains A A'; G : constrains B B' |] \
+ Goal "F Join G : Constrains A B = (Constrains F A B & Constrains G A B)";
+ by (auto_tac
+ (claset(),
+ simpset() addsimps [Constrains_def, constrains_Join]));
+ qed "Constrains_Join";
+ **)
+
+Goal "[| F : constrains A A'; G : constrains B B'; States F = States G |] \
\ ==> F Join G : constrains (A Int B) (A' Un B')";
-by (simp_tac (simpset() addsimps [constrains_Join]) 1);
+by (asm_simp_tac (simpset() addsimps [constrains_Join]) 1);
by (blast_tac (claset() addIs [constrains_weaken]) 1);
qed "constrains_Join_weaken";
-Goal "i : I ==> \
-\ (JN i:I. F i) : stable A = (ALL i:I. F i : stable A)";
+Goal "[| i : I; eqStates I F |] \
+\ ==> (JN i:I. F i) : stable A = (ALL i:I. F i : stable A)";
by (asm_simp_tac (simpset() addsimps [stable_def, constrains_JN]) 1);
qed "stable_JN";
-Goal "[| ALL i:I. F i : invariant A; i : I |] \
+Goal "[| ALL i:I. F i : invariant A; i : I; eqStates I F |] \
\ ==> (JN i:I. F i) : invariant A";
by (asm_full_simp_tac (simpset() addsimps [invariant_def, stable_JN]) 1);
by (Blast_tac 1);
bind_thm ("invariant_JN_I", ballI RS result());
-Goal "F Join G : stable A = \
-\ (F : stable A & G : stable A)";
-by (simp_tac (simpset() addsimps [stable_def, constrains_Join]) 1);
+Goal "States F = States G \
+\ ==> F Join G : stable A = (F : stable A & G : stable A)";
+by (asm_simp_tac (simpset() addsimps [stable_def, constrains_Join]) 1);
qed "stable_Join";
-Goal "[| F : invariant A; G : invariant A |] \
+Goal "[| F : invariant A; G : invariant A; States F = States G |] \
\ ==> F Join G : invariant A";
-by (full_simp_tac (simpset() addsimps [invariant_def, stable_Join]) 1);
+by (asm_full_simp_tac (simpset() addsimps [invariant_def, stable_Join]) 1);
by (Blast_tac 1);
qed "invariant_JoinI";
-Goal "i : I ==> FP (JN i:I. F i) = (INT i:I. FP (F i))";
+Goal "[| i : I; eqStates I F |] \
+\ ==> FP (JN i:I. F i) = (INT i:I. FP (F i))";
by (asm_simp_tac (simpset() addsimps [FP_def, stable_JN, INTER_def]) 1);
qed "FP_JN";
(*** Progress: transient, ensures ***)
-Goal "i : I ==> \
-\ (JN i:I. F i) : transient A = (EX i:I. F i : transient A)";
-by (auto_tac (claset(),
- simpset() addsimps [transient_def, JOIN_def]));
+
+Goal "[| (JN i:I. F i) : transient A; i: I |] ==> EX i:I. F i : transient A";
+by (full_simp_tac (simpset() addsimps [transient_def, JOIN_def, Acts_eq_raw,
+ Int_absorb1, restrict_rel_def]) 1);
+by Safe_tac;
+by (REPEAT (blast_tac (claset() addSIs [rev_bexI]) 1));
+qed "transient_JN_imp_transient";
+
+Goal "[| i : I; eqStates I F |] \
+\ ==> (JN i:I. F i) : transient A = (EX i:I. F i : transient A)";
+by (auto_tac (claset() addSIs [transient_JN_imp_transient], simpset()));
+by (auto_tac (claset(), simpset() addsimps [transient_def, Acts_JN]));
qed "transient_JN";
-Goal "F Join G : transient A = \
-\ (F : transient A | G : transient A)";
-by (auto_tac (claset(),
- simpset() addsimps [bex_Un, transient_def,
- Join_def]));
+Goal "F Join G : transient A ==> \
+\ F : transient A | G : transient A";
+by (full_simp_tac (simpset() addsimps [transient_def, Join_def, Acts_eq_raw,
+ restrict_rel_def]) 1);
+by Safe_tac;
+(*delete act:Acts F possibility*)
+by (rtac FalseE 3);
+(*delete act:Acts G possibility*)
+by (thin_tac "~ (EX act:Acts G. ?P act)" 2);
+by (REPEAT (blast_tac (claset() addSIs [rev_bexI]) 1));
+qed "transient_Join_imp_transient";
+
+Goal "States F = States G \
+\ ==> (F Join G : transient A) = (F : transient A | G : transient A)";
+by (auto_tac (claset() addSIs [transient_Join_imp_transient], simpset()));
+by (auto_tac (claset(), simpset() addsimps [transient_def, Acts_Join]));
qed "transient_Join";
-Goal "i : I ==> \
-\ (JN i:I. F i) : ensures A B = \
-\ ((ALL i:I. F i : constrains (A-B) (A Un B)) & \
-\ (EX i:I. F i : ensures A B))";
+Goal "[| i : I; eqStates I F |] \
+\ ==> (JN i:I. F i) : ensures A B = \
+\ ((ALL i:I. F i : constrains (A-B) (A Un B)) & \
+\ (EX i:I. F i : ensures A B))";
by (auto_tac (claset(),
simpset() addsimps [ensures_def, constrains_JN, transient_JN]));
qed "ensures_JN";
Goalw [ensures_def]
- "F Join G : ensures A B = \
-\ (F : constrains (A-B) (A Un B) & \
-\ G : constrains (A-B) (A Un B) & \
-\ (F : ensures A B | G : ensures A B))";
+ "States F = States G \
+\ ==> F Join G : ensures A B = \
+\ (F : constrains (A-B) (A Un B) & \
+\ G : constrains (A-B) (A Un B) & \
+\ (F : ensures A B | G : ensures A B))";
by (auto_tac (claset(),
simpset() addsimps [constrains_Join, transient_Join]));
qed "ensures_Join";
-Goalw [stable_def, constrains_def, Join_def]
- "[| F : stable A; G : constrains A A' |] \
+Goal "[| F : stable A; G : constrains A A'; \
+\ States F = States G; A <= States G |] \
\ ==> F Join G : constrains A A'";
-by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 1);
+by (forward_tac [constrains_imp_subset] 1);
+by (assume_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [stable_def, constrains_def,
+ ball_Un, Acts_Join]) 1);
by (Blast_tac 1);
qed "stable_constrains_Join";
(*Premise for G cannot use Invariant because Stable F A is weaker than
G : stable A *)
-Goal "[| F : stable A; G : invariant A |] ==> F Join G : Invariant A";
-by (full_simp_tac (simpset() addsimps [Invariant_def, invariant_def,
- Stable_eq_stable, stable_Join]) 1);
+Goal "[| F : stable A; G : invariant A; \
+\ States F = States G |] ==> F Join G : Invariant A";
+by (asm_full_simp_tac (simpset() addsimps [Invariant_def, invariant_def,
+ Stable_eq_stable, stable_Join]) 1);
by (force_tac(claset() addIs [stable_reachable, stable_Int],
simpset() addsimps [Acts_Join]) 1);
qed "stable_Join_Invariant";
-Goal "[| F : stable A; G : ensures A B |] ==> F Join G : ensures A B";
+Goal "[| F : stable A; G : ensures A B; \
+\ States F = States G |] ==> F Join G : ensures A B";
by (asm_simp_tac (simpset() addsimps [ensures_Join]) 1);
by (asm_full_simp_tac (simpset() addsimps [stable_def, ensures_def]) 1);
by (etac constrains_weaken 1);
@@ -237,35 +360,61 @@
qed "ensures_stable_Join1";
(*As above, but exchanging the roles of F and G*)
-Goal "[| F : ensures A B; G : stable A |] ==> F Join G : ensures A B";
+Goal "[| F : ensures A B; G : stable A; \
+\ States F = States G |] ==> F Join G : ensures A B";
by (stac Join_commute 1);
+by (dtac sym 1);
by (blast_tac (claset() addIs [ensures_stable_Join1]) 1);
qed "ensures_stable_Join2";
(** Diff and localTo **)
-Goalw [Join_def, Diff_def] "F Join (Diff G (Acts F)) = F Join G";
+
+Goal "States (Diff F acts) = States F";
+by (simp_tac (simpset() addsimps [Diff_def]) 1);
+qed "States_Diff";
+
+Goal "Init (Diff F acts) = Init F";
+by (auto_tac (claset() addIs [impOfSubs Init_subset_States],
+ simpset() addsimps [Diff_def]));
+qed "Init_Diff";
+
+Goal "Acts (Diff F acts) = insert (diag (States F)) (Acts F - acts)";
+by (subgoal_tac "Acts F - acts <= Pow (States F Times States F)" 1);
+by (blast_tac (claset() addEs [equalityE]
+ addDs [impOfSubs Acts_subset_Pow_States]) 2);
+by (auto_tac (claset(), simpset() addsimps [Diff_def]));
+qed "Acts_Diff";
+
+Addsimps [States_Diff, Init_Diff, Acts_Diff];
+
+
+Goalw [Join_def] "States F = States G ==> F Join (Diff G (Acts F)) = F Join G";
by (rtac program_equalityI 1);
-by Auto_tac;
+by (auto_tac (claset(), simpset() addsimps [insert_absorb]));
qed "Join_Diff2";
-Goalw [Diff_def, Disjoint_def] "Disjoint F (Diff G (Acts F))";
+Goalw [Disjoint_def] "States F = States G ==> Disjoint F (Diff G (Acts F))";
by Auto_tac;
qed "Diff_Disjoint";
+Goalw [Disjoint_def] "Disjoint F G ==> States F = States G";
+by Auto_tac;
+qed "Disjoint_States_eq";
+
Goal "[| F Join G : v localTo F; Disjoint F G |] \
\ ==> G : (INT z. stable {s. v s = z})";
by (asm_full_simp_tac
- (simpset() addsimps [localTo_def, Diff_def, Disjoint_def,
+ (simpset() addsimps [localTo_def, Disjoint_def,
Acts_Join, stable_def, constrains_def]) 1);
by (Blast_tac 1);
qed "localTo_imp_stable";
Goal "[| F Join G : v localTo F; (s,s') : act; \
-\ act : Acts G; Disjoint F G |] ==> v s' = v s";
+\ act : Acts G; Disjoint F G |] ==> v s' = v s";
by (asm_full_simp_tac
- (simpset() addsimps [localTo_def, Diff_def, Disjoint_def,
+ (simpset() addsimps [localTo_def, Disjoint_def,
Acts_Join, stable_def, constrains_def]) 1);
by (Blast_tac 1);
qed "localTo_imp_equals";
@@ -284,9 +433,15 @@
\ F Join G : w localTo F; \
\ Disjoint F G |] \
\ ==> F Join G : constrains {s. P (v s) (w s)} {s. P' (v s) (w s)}";
-by (auto_tac (claset(), simpset() addsimps [constrains_def, Acts_Join]));
+by (auto_tac (claset(),
+ simpset() addsimps [constrains_def,
+ Disjoint_States_eq RS Acts_Join]));
by (REPEAT_FIRST (dtac localTo_imp_equals THEN' REPEAT1 o atac));
-by Auto_tac;
+by (subgoal_tac "xa : States F" 1);
+by (force_tac
+ (claset() addSDs [Disjoint_States_eq,impOfSubs Acts_subset_Pow_States],
+ simpset()) 2);
+by (Force_tac 1);
qed "constrains_localTo_constrains2";
Goalw [stable_def]
@@ -316,7 +471,8 @@
by (rtac ballI 1);
by (subgoal_tac "F Join G : constrains ({s. v s = k} Int {s. v s <= w s}) \
\ ({s. v s = k} Un {s. v s <= w s})" 1);
-by (asm_simp_tac (simpset() addsimps [constrains_Join]) 2);
+by (asm_simp_tac
+ (simpset() addsimps [Disjoint_States_eq RS constrains_Join]) 2);
by (blast_tac (claset() addIs [constrains_weaken]) 2);
by (dtac (constrains_imp_Constrains RS Constrains_Int) 1 THEN etac INT_D 1);
by (etac Constrains_weaken 2);