src/HOL/UNITY/PPROD.ML

new-style infix declaration for "image"

(*  Title:      HOL/UNITY/PPROD.ML
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1999  University of Cambridge

Abstraction over replicated components (PLam)
General products of programs (Pi operation)

Probably some dead wood here!
*)


val image_eqI' = read_instantiate_sg (sign_of thy)
                     [("x", "?ff(i := ?u)")] image_eqI;

(*** Basic properties ***)

Goalw [PLam_def] "Init (PLam I F) = (INT i:I. lift_set i (Init (F i)))";
by Auto_tac;
qed "Init_PLam";

Goal "Acts (PLam I F) = insert Id (UN i:I. lift_act i `` Acts (F i))";
by (auto_tac (claset(),
	      simpset() addsimps [PLam_def]));
qed "Acts_PLam";

Addsimps [Init_PLam, Acts_PLam];

Goal "PLam {} F = SKIP";
by (simp_tac (simpset() addsimps [PLam_def]) 1);
qed "PLam_empty";

Goal "(plam i: I. SKIP) = SKIP";
by (simp_tac (simpset() addsimps [PLam_def,lift_prog_SKIP,JN_constant]) 1);
qed "PLam_SKIP";

Addsimps [PLam_SKIP, PLam_empty];

Goalw [PLam_def]
    "PLam (insert i I) F = (lift_prog i (F i)) Join (PLam I F)";
by Auto_tac;
qed "PLam_insert";

Goal "((PLam I F) <= H) = (ALL i: I. lift_prog i (F i) <= H)";
by (simp_tac (simpset() addsimps [PLam_def, JN_component_iff]) 1);
qed "PLam_component_iff";

Goalw [PLam_def] "i : I ==> lift_prog i (F i) <= (PLam I F)";
(*blast_tac doesn't use HO unification*)
by (fast_tac (claset() addIs [component_JN]) 1);
qed "component_PLam";


(** Safety & Progress **)

Goal "i : I ==>  \
\     (PLam I F : (lift_set i A) co (lift_set i B))  =  \
\     (F i : A co B)";
by (asm_simp_tac (simpset() addsimps [PLam_def, JN_constrains]) 1);
by (blast_tac (claset() addIs [lift_prog_constrains RS iffD1, 
			       constrains_imp_lift_prog_constrains]) 1);
qed "PLam_constrains";

Goal "i : I ==> (PLam I F : stable (lift_set i A)) = (F i : stable A)";
by (asm_simp_tac (simpset() addsimps [stable_def, PLam_constrains]) 1);
qed "PLam_stable";

Goal "i : I ==> \
\   PLam I F : transient A = (EX i:I. lift_prog i (F i) : transient A)";
by (asm_simp_tac (simpset() addsimps [JN_transient, PLam_def]) 1);
qed "PLam_transient";

Addsimps [PLam_constrains, PLam_stable, PLam_transient];

Goal "[| i : I;  F i : A ensures B |] ==>  \
\     PLam I F : (lift_set i A) ensures lift_set i B";
by (auto_tac (claset(), 
	      simpset() addsimps [ensures_def, lift_prog_transient_eq_disj]));
qed "PLam_ensures";

Goal "[| i : I;  F i : (A-B) co (A Un B);  F i : transient (A-B) |] ==>  \
\     PLam I F : (lift_set i A) leadsTo lift_set i B";
by (rtac (PLam_ensures RS leadsTo_Basis) 1);
by (rtac ensuresI 2);
by (ALLGOALS assume_tac);
qed "PLam_leadsTo_Basis";

Goal "[| PLam I F : AA co BB;  i: I |] \
\     ==> F i : (drop_set i AA) co (drop_set i BB)";
by (rtac lift_prog_constrains_drop_set 1);
(*rotate this assumption to be last*)
by (dres_inst_tac [("psi", "PLam I F : ?C")] asm_rl 1);
by (asm_full_simp_tac (simpset() addsimps [PLam_def, JN_constrains]) 1);
qed "PLam_constrains_drop_set";


(** invariant **)

Goal "[| F i : invariant A;  i : I |] \
\     ==> PLam I F : invariant (lift_set i A)";
by (auto_tac (claset(),
	      simpset() addsimps [invariant_def]));
qed "invariant_imp_PLam_invariant";

(*The f0 premise ensures that the product is well-defined.*)
Goal "[| PLam I F : invariant (lift_set i A);  i : I;  \
\        f0: Init (PLam I F) |] ==> F i : invariant A";
by (auto_tac (claset(),
	      simpset() addsimps [invariant_def]));
by (dres_inst_tac [("c", "f0(i:=x)")] subsetD 1);
by Auto_tac;
qed "PLam_invariant_imp_invariant";

Goal "[| i : I;  f0: Init (PLam I F) |] \
\     ==> (PLam I F : invariant (lift_set i A)) = (F i : invariant A)";
by (blast_tac (claset() addIs [invariant_imp_PLam_invariant, 
			       PLam_invariant_imp_invariant]) 1);
qed "PLam_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 \
\     ==> ((plam x:I. F) : invariant (lift_set i A)) = (F : invariant A)";
by (auto_tac (claset(),
	      simpset() addsimps [invariant_def]));
qed "const_PLam_invariant";


(** localTo **)

Goal "(PLam I F) : (sub i) localTo[C] lift_prog i (F i)";
by (simp_tac (simpset() addsimps [LOCALTO_def, Diff_def, extend_def, 
				  stable_def, constrains_def]) 1);
by (Force_tac 1);
qed "PLam_sub_localTo";


(** Reachability **)

Goal "[| f : reachable (PLam I F);  i : I |] ==> f i : reachable (F i)";
by (etac reachable.induct 1);
by (auto_tac (claset() addIs reachable.intrs, simpset()));
qed "reachable_PLam";

(*Result to justify a re-organization of this file*)
Goal "{f. ALL i:I. f i : R i} = (INT i:I. lift_set i (R i))";
by Auto_tac;
result();

Goal "reachable (PLam I F) <= (INT i:I. lift_set i (reachable (F i)))";
by (force_tac (claset() addSDs [reachable_PLam], simpset()) 1);
qed "reachable_PLam_subset1";

(*simplify using reachable_lift_prog??*)
Goal "[| i ~: I;  A : reachable (F i) |]     \
\  ==> ALL f. f : reachable (PLam I F)      \
\             --> f(i:=A) : reachable (lift_prog i (F i) Join PLam 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()) 1);
(*Init of F, action of PLam F case*)
by (res_inst_tac [("act","act")] reachable.Acts 1);
by (Force_tac 1);
by (assume_tac 1);
by (force_tac (claset() addIs [ext], simpset()) 1);
(*induction over the 2nd "reachable" assumption*)
by (eres_inst_tac [("xa","f")] reachable.induct 1);
(*Init of PLam F, action of F case*)
by (res_inst_tac [("act","lift_act i act")] reachable.Acts 1);
by (Force_tac 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 PLam I F*)
by (res_inst_tac [("act","acta")] reachable.Acts 1);
by (Force_tac 1);
by (assume_tac 1);
by (force_tac (claset() addIs [ext], simpset()) 1);
qed_spec_mp "reachable_lift_Join_PLam";


(*The index set must be finite: otherwise infinitely many copies of F can
  perform actions, and PLam can never catch up in finite time.*)
Goal "finite I \
\     ==> (INT i:I. lift_set i (reachable (F i))) <= reachable (PLam I F)";
by (etac finite_induct 1);
by (Simp_tac 1);
by (force_tac (claset() addDs [reachable_lift_Join_PLam], 
	       simpset() addsimps [PLam_insert]) 1);
qed "reachable_PLam_subset2";

Goal "finite I ==> \
\     reachable (PLam I F) = (INT i:I. lift_set i (reachable (F i)))";
by (REPEAT_FIRST (ares_tac [equalityI,
			    reachable_PLam_subset1, 
			    reachable_PLam_subset2]));
qed "reachable_PLam_eq";


(** Co **)

Goal "[| F i : A Co B;  i: I;  finite I |]  \
\     ==> PLam I F : (lift_set i A) Co (lift_set i B)";
by (auto_tac
    (claset(),
     simpset() addsimps [Constrains_def, Collect_conj_eq RS sym,
			 reachable_PLam_eq]));
by (auto_tac (claset(), 
              simpset() addsimps [constrains_def, PLam_def]));
by (REPEAT (blast_tac (claset() addIs reachable.intrs) 1));
qed "Constrains_imp_PLam_Constrains";

Goal "[| ALL j:I. f0 j : A j;   i: I |] \
\     ==> drop_set i (INT j:I. lift_set j (A j)) = A i";
by (force_tac (claset() addSIs [image_eqI'],
	       simpset() addsimps [drop_set_def]) 1);
qed "drop_set_INT_lift_set";

(*Again, we need the f0 premise so that PLam I F has an initial state;
  otherwise its Co-property is vacuous.*)
Goal "[| PLam I F : (lift_set i A) Co (lift_set i B);  \
\        i: I;  finite I;  f0: Init (PLam I F) |]  \
\     ==> F i : A Co B";
by (full_simp_tac (simpset() addsimps [Constrains_eq_constrains]) 1);
by (subgoal_tac "ALL i:I. f0 i : reachable (F i)" 1);
by (blast_tac (claset() addIs [reachable.Init]) 2);
by (dtac PLam_constrains_drop_set 1);
by (assume_tac 1);
by (asm_full_simp_tac
    (simpset() addsimps [drop_set_Int_lift_set2,
			 drop_set_INT_lift_set, reachable_PLam_eq]) 1);
qed "PLam_Constrains_imp_Constrains";


Goal "[| i: I;  finite I;  f0: Init (PLam I F) |]  \
\     ==> (PLam I F : (lift_set i A) Co (lift_set i B)) =  \
\         (F i : A Co B)";
by (blast_tac (claset() addIs [Constrains_imp_PLam_Constrains, 
			       PLam_Constrains_imp_Constrains]) 1);
qed "PLam_Constrains";

Goal "[| i: I;  finite I;  f0: Init (PLam I F) |]  \
\     ==> (PLam I F : Stable (lift_set i A)) = (F i : Stable A)";
by (asm_simp_tac (simpset() delsimps [Init_PLam]
			    addsimps [Stable_def, PLam_Constrains]) 1);
qed "PLam_Stable";


(** const_PLam (no dependence on i) doesn't require the f0 premise **)

Goal "[| (plam x:I. F) : (lift_set i A) Co (lift_set i B);  \
\        i: I;  finite I |]  \
\     ==> F : A Co B";
by (full_simp_tac (simpset() addsimps [Constrains_eq_constrains]) 1);
by (dtac PLam_constrains_drop_set 1);
by (assume_tac 1);
by (asm_full_simp_tac
    (simpset() addsimps [drop_set_INT,
			 drop_set_Int_lift_set2, Collect_conj_eq RS sym,
			 reachable_PLam_eq]) 1);
qed "const_PLam_Constrains_imp_Constrains";

Goal "[| i: I;  finite I |]  \
\     ==> ((plam x:I. F) : (lift_set i A) Co (lift_set i B)) =  \
\         (F : A Co B)";
by (blast_tac (claset() addIs [Constrains_imp_PLam_Constrains, 
			       const_PLam_Constrains_imp_Constrains]) 1);
qed "const_PLam_Constrains";

Goal "[| i: I;  finite I |]  \
\     ==> ((plam x:I. F) : Stable (lift_set i A)) = (F : Stable A)";
by (asm_simp_tac (simpset() addsimps [Stable_def, const_PLam_Constrains]) 1);
qed "const_PLam_Stable";

Goalw [Increasing_def]
     "[| i: I;  finite I |]  \
\     ==> ((plam x:I. F) : Increasing (f o sub i)) = (F : Increasing f)";
by (subgoal_tac "ALL z. {s. z <= (f o sub i) s} = lift_set i {s. z <= f s}" 1);
by (asm_simp_tac (simpset() addsimps [lift_set_sub]) 2);
by (asm_full_simp_tac
    (simpset() addsimps [finite_lessThan, const_PLam_Stable]) 1);
qed "const_PLam_Increasing";


(*** guarantees properties ***)

Goalw [PLam_def]
    "[| lift_prog i (F i): X guarantees Y;  i : I |]  \
\    ==> (PLam I F) : X guarantees Y";
by (asm_simp_tac (simpset() addsimps [guarantees_JN_I]) 1);
qed "guarantees_PLam_I";