(*  Title:      HOL/UNITY/Union.ML

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1998  University of Cambridge

Unions of programs

From Misra's Chapter 5: Asynchronous Compositions of Programs

*)

(** SKIP **)

Goal "Init SKIP = UNIV";

by (simp_tac (simpset() addsimps [SKIP_def]) 1);

qed "Init_SKIP";

Goal "Acts SKIP = {Id}";

by (simp_tac (simpset() addsimps [SKIP_def]) 1);

qed "Acts_SKIP";

Addsimps [Init_SKIP, Acts_SKIP];

Goal "reachable SKIP = UNIV";

by (force_tac (claset() addIs reachable.intrs, simpset()) 1);

qed "reachable_SKIP";

Addsimps [reachable_SKIP];

(** Join and JN **)

Goal "Init (F Join G) = Init F Int Init G";

by (simp_tac (simpset() addsimps [Join_def]) 1);

qed "Init_Join";

Goal "Acts (F Join G) = Acts F Un Acts G";

by (auto_tac (claset(), simpset() addsimps [Join_def]));

qed "Acts_Join";

Goal "Init (JN i:I. F i) = (INT i:I. Init (F i))";

by (simp_tac (simpset() addsimps [JOIN_def]) 1);

qed "Init_JN";

Goal "Acts (JN i:I. F i) = insert Id (UN i:I. Acts (F i))";

by (auto_tac (claset(), simpset() addsimps [JOIN_def]));

qed "Acts_JN";

Addsimps [Init_Join, Init_JN];

Goalw [JOIN_def, SKIP_def] "(JN i:{}. F i) = SKIP";

by Auto_tac;

qed "JN_empty";

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];

(** Algebraic laws **)

Goal "F Join G = G Join F";

by (simp_tac (simpset() addsimps [Join_def, Un_commute, Int_commute]) 1);

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);

qed "Join_assoc";

Goalw [Join_def, SKIP_def] "SKIP Join F = F";

by (rtac program_equalityI 1);

by (ALLGOALS (simp_tac (simpset() addsimps [insert_absorb])));

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])));

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;

qed "Join_absorb";

Addsimps [Join_absorb];

(*** Safety: constrains, stable, FP ***)

Goalw [constrains_def, JOIN_def]

    "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 Join G : constrains A B =  \

\     (F : constrains A B & G : constrains A B)";

by (auto_tac

    (claset(),

     simpset() addsimps [constrains_def, Join_def]));

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

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 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' |] \

\     ==> F Join G : constrains (A Int B) (A' Un B')";

by (simp_tac (simpset() addsimps [constrains_Join]) 1);

by (blast_tac (claset() addIs [constrains_weaken]) 1);

qed "constrains_Join_weaken";

Goal "I ~= {} ==> \

\     (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 "F Join G : stable A = \

\     (F : stable A & G : stable A)";

by (simp_tac (simpset() addsimps [stable_def, constrains_Join]) 1);

qed "stable_Join";

Goal "I ~= {} ==> 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 ~= {} ==> \

\   (JN i:I. F i) : transient A = (EX i:I. F i : transient A)";

by (auto_tac (claset(),

	      simpset() addsimps [transient_def, JOIN_def]));

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]));

qed "transient_Join";

Goal "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))";

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))";

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' |] \

\    ==> F Join G : constrains A A'";

by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 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);

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";

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);

by Auto_tac;

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";

by (stac Join_commute 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";

by (rtac program_equalityI 1);

by Auto_tac;

qed "Join_Diff2";

Goalw [Diff_def, Disjoint_def] "Disjoint F (Diff G (Acts F))";

by Auto_tac;

qed "Diff_Disjoint";

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,

			 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";

by (asm_full_simp_tac 

    (simpset() addsimps [localTo_def, Diff_def, Disjoint_def,

			 Acts_Join, stable_def, constrains_def]) 1);

by (Blast_tac 1);

qed "localTo_imp_equals";

Goalw [localTo_def, stable_def, constrains_def]

     "v localTo F <= (f o v) localTo F";

by (Clarify_tac 1);

by (auto_tac (claset() addSEs [allE, ballE], simpset()));

qed "localTo_imp_o_localTo";

(*** Higher-level rules involving localTo and Join ***)

Goal "[| F : constrains {s. P (v s) (w s)} {s. P' (v s) (w s)};   \

\        F Join G : v localTo F;       \

\        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 (REPEAT_FIRST (dtac localTo_imp_equals THEN' REPEAT1 o atac));

by Auto_tac;

qed "constrains_localTo_constrains2";

Goalw [stable_def]

     "[| F : stable {s. P (v s) (w s)};   \

\        F Join G : v localTo F;       \

\        F Join G : w localTo F;       \

\        Disjoint F G |]               \

\     ==> F Join G : stable {s. P (v s) (w s)}";

by (blast_tac (claset() addIs [constrains_localTo_constrains2]) 1);

qed "stable_localTo_stable2";

Goal "(UN k. {s. f s = k}) = UNIV";

by (Blast_tac 1);

qed "UN_eq_UNIV";

Goal "[| F : stable {s. v s <= w s};   \

\        F Join G : v localTo F;       \

\        F Join G : Increasing w;      \

\        Disjoint F G |]               \

\     ==> F Join G : Stable {s. v s <= w s}";

by (safe_tac (claset() addSDs [localTo_imp_stable]));

by (rewrite_goals_tac [stable_def, Stable_def, Increasing_def]);

by (subgoal_tac "ALL k: UNIV. ?H : Constrains ({s. v s = k} Int ?AA) ?AA" 1);

by (dtac ball_Constrains_UN 1);

by (full_simp_tac (simpset() addsimps [UN_eq_UNIV]) 1);

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 (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);

by Auto_tac;

qed "Increasing_localTo_Stable";