Constrains, Stable, Invariant...more of the substitution axiom, but Union
does not work well with them
(* 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
NOT CLEAR WHETHER ALL THESE FORMS ARE NEEDED:
Maybe Join instead of Un, and JOIN for UNION
*)
Goal "Init (Join prgF prgG) = Init prgF Int Init prgG";
by (simp_tac (simpset() addsimps [Join_def]) 1);
qed "Init_Join";
Goal "Acts (Join prgF prgG) = Acts prgF Un Acts prgG";
by (simp_tac (simpset() addsimps [Join_def]) 1);
qed "Acts_Join";
Goal "Init (JN i:I. prg i) = (INT i:I. Init (prg i))";
by (simp_tac (simpset() addsimps [JOIN_def]) 1);
qed "Init_JN";
Goal "Acts (JN i:I. prg i) = (UN i:I. Acts (prg i))";
by (simp_tac (simpset() addsimps [JOIN_def]) 1);
qed "Acts_JN";
Addsimps [Init_Join, Init_JN];
(** Theoretical issues **)
Goal "Join prgF prgG = Join prgG prgF";
by (simp_tac (simpset() addsimps [Join_def, Un_commute, Int_commute]) 1);
qed "Join_commute";
Goal "Join (Join prgF prgG) prgH = Join prgF (Join prgG prgH)";
by (simp_tac (simpset() addsimps [Join_def, Un_assoc, Int_assoc]) 1);
qed "Join_assoc";
(*
val field_defs = thms"program.field_defs";
val dest_defs = thms"program.dest_defs";
val dest_convs = thms"program.dest_convs";
val update_defs = thms"program.update_defs";
val make_defs = thms"program.make_defs";
val update_convs = thms"program.update_convs";
val simps = thms"program.simps";
*)
(**NOT PROVABLE because no "surjective pairing" for records
Goalw [Join_def, Null_def] "id: Acts prgF ==> Join prgF Null = prgF";
by (asm_simp_tac (simpset() addsimps [insert_absorb]) 1);
qed "Join_Null";
*)
(**NOT PROVABLE because no "surjective pairing" for records
Goalw [Join_def] "Join prgF prgF = prgF";
by Auto_tac;
qed "Join_absorb";
*)
(*** Safety: constrains, stable, FP ***)
Goalw [constrains_def, JOIN_def]
"constrains (Acts (JN i:I. prg i)) A B = \
\ (ALL i:I. constrains (Acts (prg i)) A B)";
by Auto_tac;
qed "constrains_JN";
(**FAILS, I think; see 5.4.1, Substitution Axiom.
Goalw [Constrains_def]
"Constrains (JN i:I. prg i) A B = (ALL i:I. Constrains (prg i) A B)";
by (simp_tac (simpset() addsimps [constrains_JN]) 1);
by (Blast_tac 1);
qed "Constrains_JN";
**)
Goalw [constrains_def, Join_def]
"constrains (Acts (Join prgF prgG)) A B = \
\ (constrains (Acts prgF) A B & constrains (Acts prgG) A B)";
by (simp_tac (simpset() addsimps [ball_Un]) 1);
qed "constrains_Join";
Goal "[| constrains (Acts prgF) A A'; constrains (Acts prgG) B B' |] \
\ ==> constrains (Acts (Join prgF prgG)) (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";
Goalw [stable_def]
"stable (Acts (JN i:I. prg i)) A = (ALL i:I. stable (Acts (prg i)) A)";
by (simp_tac (simpset() addsimps [constrains_JN]) 1);
qed "stable_JN";
Goal "stable (Acts (Join prgF prgG)) A = \
\ (stable (Acts prgF) A & stable (Acts prgG) A)";
by (simp_tac (simpset() addsimps [stable_def, constrains_Join]) 1);
qed "stable_Join";
Goalw [FP_def] "FP (Acts (JN i:I. prg i)) = (INT i:I. FP (Acts (prg i)))";
by (simp_tac (simpset() addsimps [stable_JN, INTER_def]) 1);
qed "FP_JN";
(*** Progress: transient, ensures ***)
Goalw [transient_def, JOIN_def]
"transient (Acts (JN i:I. prg i)) A = (EX i:I. transient (Acts (prg i)) A)";
by (Simp_tac 1);
qed "transient_JN";
Goalw [transient_def, Join_def]
"transient (Acts (Join prgF prgG)) A = \
\ (transient (Acts prgF) A | transient (Acts prgG) A)";
by (simp_tac (simpset() addsimps [bex_Un]) 1);
qed "transient_Join";
Goalw [ensures_def]
"ensures (Acts (JN i:I. prg i)) A B = \
\ ((ALL i:I. constrains (Acts (prg i)) (A-B) (A Un B)) & \
\ (EX i:I. ensures (Acts (prg i)) A B))";
by (auto_tac (claset(),
simpset() addsimps [constrains_JN, transient_JN]));
qed "ensures_JN";
Goalw [ensures_def]
"ensures (Acts (Join prgF prgG)) A B = \
\ (constrains (Acts prgF) (A-B) (A Un B) & \
\ constrains (Acts prgG) (A-B) (A Un B) & \
\ (ensures (Acts prgF) A B | ensures (Acts prgG) A B))";
by (auto_tac (claset(),
simpset() addsimps [constrains_Join, transient_Join]));
qed "ensures_Join";
Goalw [stable_def, constrains_def, Join_def]
"[| stable (Acts prgF) A; constrains (Acts prgG) A A'; id: Acts prgG |] \
\ ==> constrains (Acts (Join prgF prgG)) A A'";
by (asm_simp_tac (simpset() addsimps [ball_Un]) 1);
by (Blast_tac 1);
qed "stable_constrains_Join";
(*Premises cannot use Invariant because Stable prgF A is weaker than
stable (Acts prgG) A *)
Goal "[| stable (Acts prgF) A; Init prgG <= A; stable (Acts prgG) A |] \
\ ==> Invariant (Join prgF prgG) A";
by (simp_tac (simpset() addsimps [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 "[| stable (Acts prgF) A; ensures (Acts prgG) A B |] \
\ ==> ensures (Acts (Join prgF prgG)) 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 "[| ensures (Acts prgF) A B; stable (Acts prgG) A |] \
\ ==> ensures (Acts (Join prgF prgG)) A B";
by (stac Join_commute 1);
by (blast_tac (claset() addIs [ensures_stable_Join1]) 1);
qed "ensures_stable_Join2";