--- a/src/HOL/UNITY/PPROD.ML Fri Jan 24 14:06:49 2003 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,281 +0,0 @@
-(* 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";
-**)
-