(* 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)
Some dead wood here!
*)
(*** Basic properties ***)
Goal "Init (PLam I F) = (INT i:I. lift_set i (Init (F i)))";
by (simp_tac (simpset() addsimps [PLam_def, lift_def, lift_set_def]) 1);
qed "Init_PLam";
Addsimps [Init_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_SKIP,JN_constant]) 1);
qed "PLam_SKIP";
Addsimps [PLam_SKIP, PLam_empty];
Goalw [PLam_def] "PLam (insert i I) F = (lift i (F i)) Join (PLam I F)";
by Auto_tac;
qed "PLam_insert";
Goal "((PLam I F) <= H) = (ALL i: I. lift 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 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: but are they used anywhere? **)
Goal "[| i : I; ALL j. F j : preserves snd |] ==> \
\ (PLam I F : (lift_set i (A <*> UNIV)) co \
\ (lift_set i (B <*> UNIV))) = \
\ (F i : (A <*> UNIV) co (B <*> UNIV))";
by (asm_simp_tac (simpset() addsimps [PLam_def, JN_constrains]) 1);
by (stac (insert_Diff RS sym) 1 THEN assume_tac 1);
by (asm_simp_tac (simpset() addsimps [lift_constrains]) 1);
by (blast_tac (claset() addIs [constrains_imp_lift_constrains]) 1);
qed "PLam_constrains";
Goal "[| i : I; ALL j. F j : preserves snd |] \
\ ==> (PLam I F : stable (lift_set i (A <*> UNIV))) = \
\ (F i : stable (A <*> UNIV))";
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 i (F i) : transient A)";
by (asm_simp_tac (simpset() addsimps [JN_transient, PLam_def]) 1);
qed "PLam_transient";
(*This holds because the F j cannot change (lift_set i)*)
Goal "[| i : I; F i : (A <*> UNIV) ensures (B <*> UNIV); \
\ ALL j. F j : preserves snd |] ==> \
\ PLam I F : lift_set i (A <*> UNIV) ensures lift_set i (B <*> UNIV)";
by (auto_tac (claset(),
simpset() addsimps [ensures_def, PLam_constrains, PLam_transient,
lift_transient_eq_disj,
lift_set_Un_distrib RS sym,
lift_set_Diff_distrib RS sym,
Times_Un_distrib1 RS sym,
Times_Diff_distrib1 RS sym]));
qed "PLam_ensures";
Goal "[| i : I; \
\ F i : ((A <*> UNIV) - (B <*> UNIV)) co \
\ ((A <*> UNIV) Un (B <*> UNIV)); \
\ F i : transient ((A <*> UNIV) - (B <*> UNIV)); \
\ ALL j. F j : preserves snd |] ==> \
\ PLam I F : lift_set i (A <*> UNIV) leadsTo lift_set i (B <*> UNIV)";
by (rtac (PLam_ensures RS leadsTo_Basis) 1);
by (rtac ensuresI 2);
by (ALLGOALS assume_tac);
qed "PLam_leadsTo_Basis";
(** invariant **)
Goal "[| F i : invariant (A <*> UNIV); i : I; \
\ ALL j. F j : preserves snd |] \
\ ==> PLam I F : invariant (lift_set i (A <*> UNIV))";
by (auto_tac (claset(),
simpset() addsimps [PLam_stable, invariant_def]));
qed "invariant_imp_PLam_invariant";
Goal "ALL j. F j : preserves snd \
\ ==> (PLam I F : preserves (v o sub j o fst)) = \
\ (if j: I then F j : preserves (v o fst) else True)";
by (asm_simp_tac (simpset() addsimps [PLam_def, lift_preserves_sub]) 1);
qed "PLam_preserves_fst";
Addsimps [PLam_preserves_fst];
Goal "ALL j. F j : preserves snd ==> PLam I F : preserves snd";
by (asm_simp_tac (simpset() addsimps [PLam_def, lift_preserves_snd_I]) 1);
qed "PLam_preserves_snd";
Addsimps [PLam_preserves_snd];
AddIs [PLam_preserves_snd];
(*** guarantees properties ***)
(*This rule looks unsatisfactory because it refers to "lift". One must use
lift_guarantees_eq_lift_inv to rewrite the first subgoal and
something like lift_preserves_sub to rewrite the third. However there's
no obvious way to alternative for the third premise.*)
Goalw [PLam_def]
"[| lift i (F i): X guarantees Y; i : I; \
\ OK I (%i. lift i (F i)) |] \
\ ==> (PLam I F) : X guarantees Y";
by (asm_simp_tac (simpset() addsimps [guarantees_JN_I]) 1);
qed "guarantees_PLam_I";
Goal "Allowed (PLam I F) = (INT i:I. lift i ` Allowed(F i))";
by (simp_tac (simpset() addsimps [PLam_def]) 1);
qed "Allowed_PLam";
Addsimps [Allowed_PLam];
Goal "(PLam I F) : preserves v = (ALL i:I. F i : preserves (v o lift_map i))";
by (simp_tac (simpset() addsimps [PLam_def, lift_def, rename_preserves]) 1);
qed "PLam_preserves";
Addsimps [PLam_preserves];
(**UNUSED
(*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";
**)
(**UNUSED
(** 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??*)
Goal "[| i ~: I; A : reachable (F i) |] \
\ ==> ALL f. f : reachable (PLam I F) \
\ --> f(i:=A) : reachable (lift 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 "[| 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 "[| 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";
**)