(* Title: HOL/UNITY/PPROD.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Abstraction over replicated components (PLam)
General products of programs (Pi operation)
Some dead wood here!
*)
theory PPROD imports Lift_prog begin
definition PLam :: "[nat set, nat => ('b * ((nat=>'b) * 'c)) program]
=> ((nat=>'b) * 'c) program" where
"PLam I F == \<Squnion>i \<in> I. lift i (F i)"
syntax
"_PLam" :: "[pttrn, nat set, 'b set] => (nat => 'b) set" ("(3plam _:_./ _)" 10)
translations
"plam x : A. B" == "CONST PLam A (%x. B)"
(*** Basic properties ***)
lemma Init_PLam [simp]: "Init (PLam I F) = (\<Inter>i \<in> I. lift_set i (Init (F i)))"
by (simp add: PLam_def lift_def lift_set_def)
lemma PLam_empty [simp]: "PLam {} F = SKIP"
by (simp add: PLam_def)
lemma PLam_SKIP [simp]: "(plam i : I. SKIP) = SKIP"
by (simp add: PLam_def JN_constant)
lemma PLam_insert: "PLam (insert i I) F = (lift i (F i)) Join (PLam I F)"
by (unfold PLam_def, auto)
lemma PLam_component_iff: "((PLam I F) \<le> H) = (\<forall>i \<in> I. lift i (F i) \<le> H)"
by (simp add: PLam_def JN_component_iff)
lemma component_PLam: "i \<in> I ==> lift i (F i) \<le> (PLam I F)"
apply (unfold PLam_def)
(*blast_tac doesn't use HO unification*)
apply (fast intro: component_JN)
done
(** Safety & Progress: but are they used anywhere? **)
lemma PLam_constrains:
"[| i \<in> I; \<forall>j. F j \<in> preserves snd |]
==> (PLam I F \<in> (lift_set i (A <*> UNIV)) co
(lift_set i (B <*> UNIV))) =
(F i \<in> (A <*> UNIV) co (B <*> UNIV))"
apply (simp add: PLam_def JN_constrains)
apply (subst insert_Diff [symmetric], assumption)
apply (simp add: lift_constrains)
apply (blast intro: constrains_imp_lift_constrains)
done
lemma PLam_stable:
"[| i \<in> I; \<forall>j. F j \<in> preserves snd |]
==> (PLam I F \<in> stable (lift_set i (A <*> UNIV))) =
(F i \<in> stable (A <*> UNIV))"
by (simp add: stable_def PLam_constrains)
lemma PLam_transient:
"i \<in> I ==>
PLam I F \<in> transient A = (\<exists>i \<in> I. lift i (F i) \<in> transient A)"
by (simp add: JN_transient PLam_def)
text{*This holds because the @{term "F j"} cannot change @{term "lift_set i"}*}
lemma PLam_ensures:
"[| i \<in> I; F i \<in> (A <*> UNIV) ensures (B <*> UNIV);
\<forall>j. F j \<in> preserves snd |]
==> PLam I F \<in> lift_set i (A <*> UNIV) ensures lift_set i (B <*> UNIV)"
apply (simp add: ensures_def PLam_constrains PLam_transient
lift_set_Un_distrib [symmetric] lift_set_Diff_distrib [symmetric]
Times_Un_distrib1 [symmetric] Times_Diff_distrib1 [symmetric])
apply (rule rev_bexI, assumption)
apply (simp add: lift_transient)
done
lemma PLam_leadsTo_Basis:
"[| i \<in> I;
F i \<in> ((A <*> UNIV) - (B <*> UNIV)) co
((A <*> UNIV) \<union> (B <*> UNIV));
F i \<in> transient ((A <*> UNIV) - (B <*> UNIV));
\<forall>j. F j \<in> preserves snd |]
==> PLam I F \<in> lift_set i (A <*> UNIV) leadsTo lift_set i (B <*> UNIV)"
by (rule PLam_ensures [THEN leadsTo_Basis], rule_tac [2] ensuresI)
(** invariant **)
lemma invariant_imp_PLam_invariant:
"[| F i \<in> invariant (A <*> UNIV); i \<in> I;
\<forall>j. F j \<in> preserves snd |]
==> PLam I F \<in> invariant (lift_set i (A <*> UNIV))"
by (auto simp add: PLam_stable invariant_def)
lemma PLam_preserves_fst [simp]:
"\<forall>j. F j \<in> preserves snd
==> (PLam I F \<in> preserves (v o sub j o fst)) =
(if j \<in> I then F j \<in> preserves (v o fst) else True)"
by (simp add: PLam_def lift_preserves_sub)
lemma PLam_preserves_snd [simp,intro]:
"\<forall>j. F j \<in> preserves snd ==> PLam I F \<in> preserves snd"
by (simp add: PLam_def lift_preserves_snd_I)
(*** guarantees properties ***)
text{*This rule looks unsatisfactory because it refers to @{term lift}.
One must use
@{text lift_guarantees_eq_lift_inv} to rewrite the first subgoal and
something like @{text lift_preserves_sub} to rewrite the third. However
there's no obvious alternative for the third premise.*}
lemma guarantees_PLam_I:
"[| lift i (F i): X guarantees Y; i \<in> I;
OK I (%i. lift i (F i)) |]
==> (PLam I F) \<in> X guarantees Y"
apply (unfold PLam_def)
apply (simp add: guarantees_JN_I)
done
lemma Allowed_PLam [simp]:
"Allowed (PLam I F) = (\<Inter>i \<in> I. lift i ` Allowed(F i))"
by (simp add: PLam_def)
lemma PLam_preserves [simp]:
"(PLam I F) \<in> preserves v = (\<forall>i \<in> I. F i \<in> preserves (v o lift_map i))"
by (simp add: PLam_def lift_def rename_preserves)
(**UNUSED
(*The f0 premise ensures that the product is well-defined.*)
lemma PLam_invariant_imp_invariant:
"[| PLam I F \<in> invariant (lift_set i A); i \<in> I;
f0: Init (PLam I F) |] ==> F i \<in> invariant A"
apply (auto simp add: invariant_def)
apply (drule_tac c = "f0 (i:=x) " in subsetD)
apply auto
done
lemma PLam_invariant:
"[| i \<in> I; f0: Init (PLam I F) |]
==> (PLam I F \<in> invariant (lift_set i A)) = (F i \<in> invariant A)"
apply (blast intro: invariant_imp_PLam_invariant PLam_invariant_imp_invariant)
done
(*The f0 premise isn't needed if F is a constant program because then
we get an initial state by replicating that of F*)
lemma reachable_PLam:
"i \<in> I
==> ((plam x \<in> I. F) \<in> invariant (lift_set i A)) = (F \<in> invariant A)"
apply (auto simp add: invariant_def)
done
**)
(**UNUSED
(** Reachability **)
Goal "[| f \<in> reachable (PLam I F); i \<in> I |] ==> f i \<in> reachable (F i)"
apply (erule reachable.induct)
apply (auto intro: reachable.intrs)
done
(*Result to justify a re-organization of this file*)
lemma "{f. \<forall>i \<in> I. f i \<in> R i} = (\<Inter>i \<in> I. lift_set i (R i))"
by auto
lemma reachable_PLam_subset1:
"reachable (PLam I F) \<subseteq> (\<Inter>i \<in> I. lift_set i (reachable (F i)))"
apply (force dest!: reachable_PLam)
done
(*simplify using reachable_lift??*)
lemma reachable_lift_Join_PLam [rule_format]:
"[| i \<notin> I; A \<in> reachable (F i) |]
==> \<forall>f. f \<in> reachable (PLam I F)
--> f(i:=A) \<in> reachable (lift i (F i) Join PLam I F)"
apply (erule reachable.induct)
apply (ALLGOALS Clarify_tac)
apply (erule reachable.induct)
(*Init, Init case*)
apply (force intro: reachable.intrs)
(*Init of F, action of PLam F case*)
apply (rule_tac act = act in reachable.Acts)
apply force
apply assumption
apply (force intro: ext)
(*induction over the 2nd "reachable" assumption*)
apply (erule_tac xa = f in reachable.induct)
(*Init of PLam F, action of F case*)
apply (rule_tac act = "lift_act i act" in reachable.Acts)
apply force
apply (force intro: reachable.Init)
apply (force intro: ext simp add: lift_act_def)
(*last case: an action of PLam I F*)
apply (rule_tac act = acta in reachable.Acts)
apply force
apply assumption
apply (force intro: ext)
done
(*The index set must be finite: otherwise infinitely many copies of F can
perform actions, and PLam can never catch up in finite time.*)
lemma reachable_PLam_subset2:
"finite I
==> (\<Inter>i \<in> I. lift_set i (reachable (F i))) \<subseteq> reachable (PLam I F)"
apply (erule finite_induct)
apply (simp (no_asm))
apply (force dest: reachable_lift_Join_PLam simp add: PLam_insert)
done
lemma reachable_PLam_eq:
"finite I ==>
reachable (PLam I F) = (\<Inter>i \<in> I. lift_set i (reachable (F i)))"
apply (REPEAT_FIRST (ares_tac [equalityI, reachable_PLam_subset1, reachable_PLam_subset2]))
done
(** Co **)
lemma Constrains_imp_PLam_Constrains:
"[| F i \<in> A Co B; i \<in> I; finite I |]
==> PLam I F \<in> (lift_set i A) Co (lift_set i B)"
apply (auto simp add: Constrains_def Collect_conj_eq [symmetric] reachable_PLam_eq)
apply (auto simp add: constrains_def PLam_def)
apply (REPEAT (blast intro: reachable.intrs))
done
lemma PLam_Constrains:
"[| i \<in> I; finite I; f0: Init (PLam I F) |]
==> (PLam I F \<in> (lift_set i A) Co (lift_set i B)) =
(F i \<in> A Co B)"
apply (blast intro: Constrains_imp_PLam_Constrains PLam_Constrains_imp_Constrains)
done
lemma PLam_Stable:
"[| i \<in> I; finite I; f0: Init (PLam I F) |]
==> (PLam I F \<in> Stable (lift_set i A)) = (F i \<in> Stable A)"
apply (simp del: Init_PLam add: Stable_def PLam_Constrains)
done
(** const_PLam (no dependence on i) doesn't require the f0 premise **)
lemma const_PLam_Constrains:
"[| i \<in> I; finite I |]
==> ((plam x \<in> I. F) \<in> (lift_set i A) Co (lift_set i B)) =
(F \<in> A Co B)"
apply (blast intro: Constrains_imp_PLam_Constrains const_PLam_Constrains_imp_Constrains)
done
lemma const_PLam_Stable:
"[| i \<in> I; finite I |]
==> ((plam x \<in> I. F) \<in> Stable (lift_set i A)) = (F \<in> Stable A)"
apply (simp add: Stable_def const_PLam_Constrains)
done
lemma const_PLam_Increasing:
"[| i \<in> I; finite I |]
==> ((plam x \<in> I. F) \<in> Increasing (f o sub i)) = (F \<in> Increasing f)"
apply (unfold Increasing_def)
apply (subgoal_tac "\<forall>z. {s. z \<subseteq> (f o sub i) s} = lift_set i {s. z \<subseteq> f s}")
apply (asm_simp_tac (simpset () add: lift_set_sub) 2)
apply (simp add: finite_lessThan const_PLam_Stable)
done
**)
end