UNITY fully working at last...
(* Title: HOL/UNITY/PPROD.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
*)
val rinst = read_instantiate_sg (sign_of thy);
(**** PPROD ****)
(*** Basic properties ***)
Goalw [lift_set_def] "(f : lift_set i A) = (f i : A)";
by Auto_tac;
qed "lift_set_iff";
AddIffs [lift_set_iff];
Goalw [lift_act_def] "lift_act i Id = Id";
by Auto_tac;
qed "lift_act_Id";
Addsimps [lift_act_Id];
(*NEEDED????????????????????????????????????????????????????????????????*)
Goal "Id : lift_act i `` Acts F";
by (auto_tac (claset() addSIs [lift_act_Id RS sym],
simpset() addsimps [image_iff]));
qed "Id_mem_lift_act";
Goalw [lift_prog_def] "Init (lift_prog i F) = lift_set i (Init F)";
by Auto_tac;
qed "Init_lift_prog";
Addsimps [Init_lift_prog];
Goalw [lift_prog_def] "Acts (lift_prog i F) = lift_act i `` Acts F";
by (auto_tac (claset() addIs [Id_in_Acts RSN (2,image_eqI)], simpset()));
qed "Acts_lift_prog";
Goalw [PPROD_def] "Init (PPROD I F) = (INT i:I. lift_set i (Init (F i)))";
by Auto_tac;
qed "Init_PPROD";
Addsimps [Init_PPROD];
Goalw [lift_act_def]
"((f,f') : lift_act i act) = (EX s'. f' = f(i := s') & (f i, s') : act)";
by (Blast_tac 1);
qed "lift_act_eq";
AddIffs [lift_act_eq];
Goal "Acts (PPROD I F) = insert Id (UN i:I. lift_act i `` Acts (F i))";
by (auto_tac (claset(),
simpset() addsimps [PPROD_def, Acts_lift_prog, Acts_JN]));
qed "Acts_PPROD";
Goal "PPROD {} F = SKIP";
by (simp_tac (simpset() addsimps [PPROD_def]) 1);
qed "PPROD_empty";
Goal "(PPI i: I. SKIP) = SKIP";
by (auto_tac (claset() addSIs [program_equalityI],
simpset() addsimps [Acts_lift_prog, SKIP_def, Acts_PPROD]));
qed "PPROD_SKIP";
Addsimps [PPROD_SKIP, PPROD_empty];
Goalw [PPROD_def]
"PPROD (insert i I) F = (lift_prog i (F i)) Join (PPROD I F)";
by Auto_tac;
qed "PPROD_insert";
Goalw [PPROD_def] "i : I ==> component (lift_prog i (F i)) (PPROD I F)";
(*blast_tac doesn't use HO unification*)
by (fast_tac (claset() addIs [component_JN]) 1);
qed "component_PPROD";
(*** Safety: constrains, stable, invariant ***)
(** 1st formulation of lifting **)
Goal "(lift_prog i F : constrains (lift_set i A) (lift_set i B)) = \
\ (F : constrains A B)";
by (auto_tac (claset(),
simpset() addsimps [constrains_def, Acts_lift_prog]));
by (Blast_tac 2);
by (Force_tac 1);
qed "lift_prog_constrains_eq";
Goal "(lift_prog i F : stable (lift_set i A)) = (F : stable A)";
by (simp_tac (simpset() addsimps [stable_def, lift_prog_constrains_eq]) 1);
qed "lift_prog_stable_eq";
(*This one looks strange! Proof probably is by case analysis on i=j.*)
Goal "F i : constrains A B \
\ ==> lift_prog j (F j) : constrains (lift_set i A) (lift_set i B)";
by (auto_tac (claset(),
simpset() addsimps [constrains_def, Acts_lift_prog]));
by (REPEAT (Blast_tac 1));
qed "constrains_imp_lift_prog_constrains";
Goal "i : I ==> \
\ (PPROD I F : constrains (lift_set i A) (lift_set i B)) = \
\ (F i : constrains A B)";
by (asm_simp_tac (simpset() addsimps [PPROD_def, constrains_JN]) 1);
by (blast_tac (claset() addIs [lift_prog_constrains_eq RS iffD1,
constrains_imp_lift_prog_constrains]) 1);
qed "PPROD_constrains";
Goal "i : I ==> (PPROD I F : stable (lift_set i A)) = (F i : stable A)";
by (asm_simp_tac (simpset() addsimps [stable_def, PPROD_constrains]) 1);
qed "PPROD_stable";
(** 2nd formulation of lifting **)
Goal "[| lift_prog i F : constrains AA BB |] \
\ ==> F : constrains (Applyall AA i) (Applyall BB i)";
by (auto_tac (claset(),
simpset() addsimps [Applyall_def, constrains_def,
Acts_lift_prog]));
by (force_tac (claset() addSIs [rinst [("x", "?ff(i := ?u)")] image_eqI],
simpset()) 1);
qed "lift_prog_constrains_projection";
Goal "[| PPROD I F : constrains AA BB; i: I |] \
\ ==> F i : constrains (Applyall AA i) (Applyall BB i)";
by (rtac lift_prog_constrains_projection 1);
(*rotate this assumption to be last*)
by (dres_inst_tac [("psi", "PPROD I F : ?C")] asm_rl 1);
by (asm_full_simp_tac (simpset() addsimps [PPROD_def, constrains_JN]) 1);
qed "PPROD_constrains_projection";
(** invariant **)
(*????????????????*)
Goal "F : invariant A ==> lift_prog i F : invariant (lift_set i A)";
by (auto_tac (claset(),
simpset() addsimps [invariant_def, lift_prog_stable_eq]));
qed "invariant_imp_lift_prog_invariant";
(*????????????????*)
Goal "[| lift_prog i F : invariant (lift_set i A) |] ==> F : invariant A";
by (auto_tac (claset(),
simpset() addsimps [invariant_def, lift_prog_stable_eq]));
qed "lift_prog_invariant_imp_invariant";
(*NOT clear that the previous lemmas help in proving this one.*)
Goal "[| F i : invariant A; i : I |] \
\ ==> PPROD I F : invariant (lift_set i A)";
by (auto_tac (claset(),
simpset() addsimps [invariant_def, PPROD_stable]));
qed "invariant_imp_PPROD_invariant";
(*The f0 premise ensures that the product is well-defined.*)
Goal "[| PPROD I F : invariant (lift_set i A); i : I; \
\ f0: Init (PPROD I F) |] ==> F i : invariant A";
by (auto_tac (claset(),
simpset() addsimps [invariant_def, PPROD_stable]));
by (dres_inst_tac [("c", "f0(i:=x)")] subsetD 1);
by Auto_tac;
qed "PPROD_invariant_imp_invariant";
Goal "[| i : I; f0: Init (PPROD I F) |] \
\ ==> (PPROD I F : invariant (lift_set i A)) = (F i : invariant A)";
by (blast_tac (claset() addIs [invariant_imp_PPROD_invariant,
PPROD_invariant_imp_invariant]) 1);
qed "PPROD_invariant";
(*The f0 premise isn't needed if F is a constant program because then
we get an initial state by replicating that of F*)
Goal "i : I \
\ ==> ((PPI x:I. F) : invariant (lift_set i A)) = (F : invariant A)";
by (auto_tac (claset(),
simpset() addsimps [invariant_def, PPROD_stable]));
qed "PFUN_invariant";
(*** Substitution Axiom versions: Constrains, Stable ***)
(** Reachability **)
Goal "[| f : reachable (PPROD I F); i : I |] ==> f i : reachable (F i)";
by (etac reachable.induct 1);
by (auto_tac
(claset() addIs reachable.intrs,
simpset() addsimps [Acts_PPROD]));
qed "reachable_PPROD";
Goal "reachable (PPROD I F) <= {f. ALL i:I. f i : reachable (F i)}";
by (force_tac (claset() addSDs [reachable_PPROD], simpset()) 1);
qed "reachable_PPROD_subset1";
Goal "[| i ~: I; A : reachable (F i) |] \
\ ==> ALL f. f : reachable (PPROD I F) \
\ --> f(i:=A) : reachable (lift_prog i (F i) Join PPROD I F)";
by (etac reachable.induct 1);
by (ALLGOALS Clarify_tac);
by (etac reachable.induct 1);
(*Init, Init case*)
by (force_tac (claset() addIs reachable.intrs,
simpset() addsimps [Acts_lift_prog]) 1);
(*Init of F, action of PPROD F case*)
by (rtac reachable.Acts 1);
by (force_tac (claset(), simpset() addsimps [Acts_Join]) 1);
by (assume_tac 1);
by (force_tac (claset() addIs [ext], simpset() addsimps [Acts_PPROD]) 1);
(*induction over the 2nd "reachable" assumption*)
by (eres_inst_tac [("xa","f")] reachable.induct 1);
(*Init of PPROD F, action of F case*)
by (res_inst_tac [("act","lift_act i act")] reachable.Acts 1);
by (force_tac (claset(), simpset() addsimps [Acts_lift_prog, Acts_Join]) 1);
by (force_tac (claset() addIs [reachable.Init], simpset()) 1);
by (force_tac (claset() addIs [ext], simpset() addsimps [lift_act_def]) 1);
(*last case: an action of PPROD I F*)
by (rtac reachable.Acts 1);
by (force_tac (claset(), simpset() addsimps [Acts_Join]) 1);
by (assume_tac 1);
by (force_tac (claset() addIs [ext], simpset() addsimps [Acts_PPROD]) 1);
qed_spec_mp "reachable_lift_Join_PPROD";
(*The index set must be finite: otherwise infinitely many copies of F can
perform actions, and PPROD can never catch up in finite time.*)
Goal "finite I \
\ ==> {f. ALL i:I. f i : reachable (F i)} <= reachable (PPROD I F)";
by (etac finite_induct 1);
by (Simp_tac 1);
by (force_tac (claset() addDs [reachable_lift_Join_PPROD],
simpset() addsimps [PPROD_insert]) 1);
qed "reachable_PPROD_subset2";
Goal "finite I ==> \
\ reachable (PPROD I F) = {f. ALL i:I. f i : reachable (F i)}";
by (REPEAT_FIRST (ares_tac [equalityI,
reachable_PPROD_subset1,
reachable_PPROD_subset2]));
qed "reachable_PPROD_eq";
(** Constrains **)
Goal "[| F i : Constrains A B; i: I; finite I |] \
\ ==> PPROD I F : Constrains (lift_set i A) (lift_set i B)";
by (auto_tac
(claset(),
simpset() addsimps [Constrains_def, Collect_conj_eq RS sym,
reachable_PPROD_eq]));
by (auto_tac (claset(),
simpset() addsimps [constrains_def, Acts_lift_prog, PPROD_def,
Acts_JN]));
by (REPEAT (blast_tac (claset() addIs reachable.intrs) 1));
qed "Constrains_imp_PPROD_Constrains";
Goal "[| ALL i:I. f0 i : R i; i: I |] \
\ ==> Applyall ({f. (ALL i:I. f i : R i)} Int lift_set i A) i = R i Int A";
by (force_tac (claset() addSIs [rinst [("x", "?ff(i := ?u)")] image_eqI],
simpset() addsimps [Applyall_def, lift_set_def]) 1);
qed "Applyall_Int_depend";
(*Again, we need the f0 premise because otherwise Constrains holds trivially
for PPROD I F.*)
Goal "[| PPROD I F : Constrains (lift_set i A) (lift_set i B); \
\ i: I; finite I; f0: Init (PPROD I F) |] \
\ ==> F i : Constrains A B";
by (full_simp_tac (simpset() addsimps [Constrains_def]) 1);
by (subgoal_tac "ALL i:I. f0 i : reachable (F i)" 1);
by (blast_tac (claset() addIs [reachable.Init]) 2);
by (dtac PPROD_constrains_projection 1);
by (assume_tac 1);
by (asm_full_simp_tac
(simpset() addsimps [Applyall_Int_depend, reachable_PPROD_eq]) 1);
qed "PPROD_Constrains_imp_Constrains";
Goal "[| i: I; finite I; f0: Init (PPROD I F) |] \
\ ==> (PPROD I F : Constrains (lift_set i A) (lift_set i B)) = \
\ (F i : Constrains A B)";
by (blast_tac (claset() addIs [Constrains_imp_PPROD_Constrains,
PPROD_Constrains_imp_Constrains]) 1);
qed "PPROD_Constrains";
Goal "[| i: I; finite I; f0: Init (PPROD I F) |] \
\ ==> (PPROD I F : Stable (lift_set i A)) = (F i : Stable A)";
by (asm_simp_tac (simpset() delsimps [Init_PPROD]
addsimps [Stable_def, PPROD_Constrains]) 1);
qed "PPROD_Stable";
(** PFUN (no dependence on i) doesn't require the f0 premise **)
Goal "i: I \
\ ==> Applyall ({f. (ALL i:I. f i : R)} Int lift_set i A) i = R Int A";
by (force_tac (claset(), simpset() addsimps [Applyall_def]) 1);
qed "Applyall_Int";
Goal "[| (PPI x:I. F) : Constrains (lift_set i A) (lift_set i B); \
\ i: I; finite I |] \
\ ==> F : Constrains A B";
by (full_simp_tac (simpset() addsimps [Constrains_def]) 1);
by (dtac PPROD_constrains_projection 1);
by (assume_tac 1);
by (asm_full_simp_tac
(simpset() addsimps [Applyall_Int, Collect_conj_eq RS sym,
reachable_PPROD_eq]) 1);
qed "PFUN_Constrains_imp_Constrains";
Goal "[| i: I; finite I |] \
\ ==> ((PPI x:I. F) : Constrains (lift_set i A) (lift_set i B)) = \
\ (F : Constrains A B)";
by (blast_tac (claset() addIs [Constrains_imp_PPROD_Constrains,
PFUN_Constrains_imp_Constrains]) 1);
qed "PFUN_Constrains";
Goal "[| i: I; finite I |] \
\ ==> ((PPI x:I. F) : Stable (lift_set i A)) = (F : Stable A)";
by (asm_simp_tac (simpset() addsimps [Stable_def, PFUN_Constrains]) 1);
qed "PFUN_Stable";
(*** guarantees properties ***)
Goal "drop_act i (lift_act i act) = act";
by (force_tac (claset() addSIs [rinst [("x", "?ff(i := ?u)")] exI],
simpset() addsimps [drop_act_def, lift_act_def]) 1);
qed "lift_act_inverse";
Addsimps [lift_act_inverse];
Goal "(lift_prog i F) Join G = lift_prog i H \
\ ==> EX J. H = F Join J";
by (etac program_equalityE 1);
by (auto_tac (claset(), simpset() addsimps [Acts_lift_prog, Acts_Join]));
by (res_inst_tac [("x",
"mk_program(Applyall(Init G) i, drop_act i `` Acts G)")]
exI 1);
by (rtac program_equalityI 1);
(*Init*)
by (simp_tac (simpset() addsimps [Applyall_def]) 1);
(*Blast_tac can't do HO unification, needed to invent function states*)
by (fast_tac (claset() addEs [equalityE]) 1);
(*Now for the Actions*)
by (dres_inst_tac [("f", "op `` (drop_act i)")] arg_cong 1);
by (asm_full_simp_tac
(simpset() addsimps [insert_absorb, Acts_Join,
image_Un, image_compose RS sym, o_def]) 1);
qed "lift_prog_Join_eq_lift_prog_D";
Goal "F : X guarantees Y \
\ ==> lift_prog i F : (lift_prog i `` X) guarantees (lift_prog i `` Y)";
by (rtac guaranteesI 1);
by Auto_tac;
by (blast_tac (claset() addDs [lift_prog_Join_eq_lift_prog_D, guaranteesD]) 1);
qed "lift_prog_guarantees";