--- a/src/HOL/UNITY/PPROD.ML Fri Aug 06 17:28:45 1999 +0200
+++ b/src/HOL/UNITY/PPROD.ML Fri Aug 06 17:29:18 1999 +0200
@@ -1,183 +1,25 @@
(* Title: HOL/UNITY/PPROD.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1998 University of Cambridge
+ Copyright 1999 University of Cambridge
+
+Abstraction over replicated components (PLam)
+General products of programs (Pi operation)
+
+It is not clear that any of this is good for anything.
*)
val rinst = read_instantiate_sg (sign_of thy);
-(**** PPROD ****)
(*** Basic properties ***)
-(** lift_set and drop_set **)
-
-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_set_def] "lift_set i (A Int B) = lift_set i A Int lift_set i B";
-by Auto_tac;
-qed "lift_set_Int";
-
-(*USED?? converse fails*)
-Goalw [drop_set_def] "f : A ==> f i : drop_set i A";
-by Auto_tac;
-qed "drop_set_I";
-
-(** lift_act and drop_act **)
-
-Goalw [lift_act_def] "lift_act i Id = Id";
-by Auto_tac;
-by (etac rev_mp 1);
-by (dtac sym 1);
-by (asm_simp_tac (simpset() addsimps [fun_upd_idem_iff]) 1);
-qed "lift_act_Id";
-Addsimps [lift_act_Id];
-
-Goalw [drop_act_def] "(s, s') : act ==> (s i, s' i) : drop_act i act";
-by (auto_tac (claset() addSIs [image_eqI], simpset()));
-qed "drop_act_I";
-
-Goalw [drop_act_def] "drop_act i Id = Id";
-by Auto_tac;
-by (res_inst_tac [("x", "((%u. x), (%u. x))")] image_eqI 1);
-by Auto_tac;
-qed "drop_act_Id";
-Addsimps [drop_act_Id];
-
-(** lift_prog and drop_prog **)
-
-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";
-Addsimps [Acts_lift_prog];
-
-Goalw [drop_prog_def] "Init (drop_prog i F) = drop_set i (Init F)";
-by Auto_tac;
-qed "Init_drop_prog";
-Addsimps [Init_drop_prog];
-
-Goalw [drop_prog_def] "Acts (drop_prog i F) = drop_act i `` Acts F";
-by (auto_tac (claset() addIs [Id_in_Acts RSN (2,image_eqI)], simpset()));
-qed "Acts_drop_prog";
-Addsimps [Acts_drop_prog];
-
-Goal "F component G ==> lift_prog i F component lift_prog i G";
-by (full_simp_tac (simpset() addsimps [component_eq_subset]) 1);
-by Auto_tac;
-qed "lift_prog_mono";
-
-Goal "F component G ==> drop_prog i F component drop_prog i G";
-by (full_simp_tac (simpset() addsimps [component_eq_subset, drop_set_def]) 1);
-by Auto_tac;
-qed "drop_prog_mono";
-
-Goal "lift_prog i SKIP = SKIP";
-by (auto_tac (claset() addSIs [program_equalityI],
- simpset() addsimps [SKIP_def, lift_prog_def]));
-qed "lift_prog_SKIP";
-
-Goal "lift_prog i (F Join G) = (lift_prog i F) Join (lift_prog i G)";
-by (rtac program_equalityI 1);
-by (auto_tac (claset(), simpset() addsimps [Acts_Join]));
-qed "lift_prog_Join";
-
-Goal "lift_prog i (JOIN I F) = (JN j:I. lift_prog i (F j))";
-by (rtac program_equalityI 1);
-by (auto_tac (claset(), simpset() addsimps [Acts_JN]));
-qed "lift_prog_JN";
-
-Goal "drop_prog i SKIP = SKIP";
-by (auto_tac (claset() addSIs [program_equalityI],
- simpset() addsimps [SKIP_def, drop_set_def, drop_prog_def]));
-qed "drop_prog_SKIP";
-
-
-(** Injectivity of lift_set, lift_act, lift_prog **)
-
-Goalw [lift_set_def, drop_set_def] "drop_set i (lift_set i F) = F";
-by Auto_tac;
-qed "lift_set_inverse";
-Addsimps [lift_set_inverse];
-
-Goal "inj (lift_set i)";
-by (rtac inj_on_inverseI 1);
-by (rtac lift_set_inverse 1);
-qed "inj_lift_set";
-
-(*Because A and B could differ outside i, cannot generalize result to
- drop_set i (A Int B) = drop_set i A Int drop_set i B
-*)
-Goalw [lift_set_def, drop_set_def]
- "drop_set i ((lift_set i A) Int B) = A Int (drop_set i B)";
-by Auto_tac;
-qed "drop_set_lift_set_Int";
-
-Goalw [lift_set_def, drop_set_def]
- "drop_set i (B Int (lift_set i A)) = (drop_set i B) Int A";
-by Auto_tac;
-qed "drop_set_lift_set_Int2";
-
-Goalw [drop_set_def]
- "i : I ==> drop_set i (INT j:I. lift_set j A) = A";
-by Auto_tac;
-qed "drop_set_INT";
-
-Goalw [lift_act_def, drop_act_def] "drop_act i (lift_act i act) = act";
-by Auto_tac;
-by (res_inst_tac [("x", "f(i:=a)")] exI 1);
-by (Asm_simp_tac 1);
-by (res_inst_tac [("x", "f(i:=b)")] exI 1);
-by (Asm_simp_tac 1);
-by (rtac ext 1);
-by (Asm_simp_tac 1);
-qed "lift_act_inverse";
-Addsimps [lift_act_inverse];
-
-Goal "inj (lift_act i)";
-by (rtac inj_on_inverseI 1);
-by (rtac lift_act_inverse 1);
-qed "inj_lift_act";
-
-Goal "drop_prog i (lift_prog i F) = F";
-by (simp_tac (simpset() addsimps [lift_prog_def, drop_prog_def]) 1);
-by (rtac program_equalityI 1);
-by (simp_tac (simpset() addsimps [image_compose RS sym, o_def]) 2);
-by (Simp_tac 1);
-qed "lift_prog_inverse";
-Addsimps [lift_prog_inverse];
-
-Goal "inj (lift_prog i)";
-by (rtac inj_on_inverseI 1);
-by (rtac lift_prog_inverse 1);
-qed "inj_lift_prog";
-
-
-(** PLam **)
-
Goalw [PLam_def] "Init (PLam I F) = (INT i:I. lift_set i (Init (F i)))";
by Auto_tac;
qed "Init_PLam";
Addsimps [Init_PLam];
-(*For compatibility with the original definition and perhaps simpler proofs*)
-Goalw [lift_act_def]
- "((f,f') : lift_act i act) = (EX s'. f' = f(i := s') & (f i, s') : act)";
-by Auto_tac;
-by (rtac exI 1);
-by Auto_tac;
-qed "lift_act_eq";
-AddIffs [lift_act_eq];
-
-
Goal "Acts (PLam I F) = insert Id (UN i:I. lift_act i `` Acts (F i))";
by (auto_tac (claset(),
simpset() addsimps [PLam_def, Acts_JN]));
@@ -204,51 +46,7 @@
qed "component_PLam";
-(*** Safety: co, stable, invariant ***)
-
-(** Safety and lift_prog **)
-
-Goal "(lift_prog i F : (lift_set i A) co (lift_set i B)) = \
-\ (F : A co B)";
-by (auto_tac (claset(),
- simpset() addsimps [constrains_def]));
-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.
- If i~=j then lift_prog j (F j) does nothing to lift_set i, and the
- premise ensures A<=B.*)
-Goal "F i : A co B \
-\ ==> lift_prog j (F j) : (lift_set i A) co (lift_set i B)";
-by (auto_tac (claset(),
- simpset() addsimps [constrains_def]));
-by (REPEAT (Blast_tac 1));
-qed "constrains_imp_lift_prog_constrains";
-
-(** safety properties for program unit j hold trivially of unit i **)
-Goalw [constrains_def]
- "[| i~=j; A<= B|] ==> lift_prog i F : (lift_set j A) co (lift_set j B)";
-by Auto_tac;
-qed "neq_imp_lift_prog_constrains";
-
-Goalw [stable_def]
- "i~=j ==> lift_prog i F : stable (lift_set j A)";
-by (blast_tac (claset() addIs [neq_imp_lift_prog_constrains]) 1);
-qed "neq_imp_lift_prog_stable";
-
-bind_thm ("neq_imp_lift_prog_Constrains",
- neq_imp_lift_prog_constrains RS constrains_imp_Constrains);
-
-bind_thm ("neq_imp_lift_prog_Stable",
- neq_imp_lift_prog_stable RS stable_imp_Stable);
-
-
-(** Safety and PLam **)
+(** Safety **)
Goal "i : I ==> \
\ (PLam I F : (lift_set i A) co (lift_set i B)) = \
@@ -262,33 +60,26 @@
by (asm_simp_tac (simpset() addsimps [stable_def, PLam_constrains]) 1);
qed "PLam_stable";
-
-(** Safety, lift_prog + drop_set **)
-
+(*Possibly useful in Lift_set.ML?*)
Goal "[| lift_prog i F : AA co BB |] \
\ ==> F : (drop_set i AA) co (drop_set i BB)";
by (auto_tac (claset(),
simpset() addsimps [constrains_def, drop_set_def]));
by (force_tac (claset() addSIs [rinst [("x", "?ff(i := ?u)")] image_eqI],
simpset()) 1);
-qed "lift_prog_constrains_projection";
+qed "lift_prog_constrains_drop_set";
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_projection 1);
+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, constrains_JN]) 1);
-qed "PLam_constrains_projection";
+qed "PLam_constrains_drop_set";
(** 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_eq";
-
Goal "[| F i : invariant A; i : I |] \
\ ==> PLam I F : invariant (lift_set i A)";
by (auto_tac (claset(),
@@ -319,46 +110,7 @@
qed "const_PLam_invariant";
-(*** Substitution Axiom versions: Co, Stable ***)
-
-(*** Reachability ***)
-
-(** for lift_prog **)
-
-Goal "s : reachable F ==> f(i:=s) : reachable (lift_prog i F)";
-by (etac reachable.induct 1);
-by (force_tac (claset() addIs [reachable.Acts, ext],
- simpset()) 2);
-by (force_tac (claset() addIs [reachable.Init], simpset()) 1);
-qed "reachable_lift_progI";
-
-Goal "f : reachable (lift_prog i F) ==> f i : reachable F";
-by (etac reachable.induct 1);
-by Auto_tac;
-by (ALLGOALS (blast_tac (claset() addIs reachable.intrs)));
-qed "reachable_lift_progD";
-
-Goal "reachable (lift_prog i F) = lift_set i (reachable F)";
-by Auto_tac;
-by (etac reachable_lift_progD 1);
-ren "f" 1;
-by (dres_inst_tac [("f","f"),("i","i")] reachable_lift_progI 1);
-by Auto_tac;
-qed "reachable_lift_prog";
-
-Goal "(lift_prog i F : (lift_set i A) Co (lift_set i B)) = \
-\ (F : A Co B)";
-by (simp_tac (simpset() addsimps [Constrains_def, reachable_lift_prog,
- lift_set_Int RS sym,
- lift_prog_constrains_eq]) 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";
-
-
-(** Reachability for PLam **)
+(** Reachability **)
Goal "[| f : reachable (PLam I F); i : I |] ==> f i : reachable (F i)";
by (etac reachable.induct 1);
@@ -450,7 +202,7 @@
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_projection 1);
+by (dtac PLam_constrains_drop_set 1);
by (assume_tac 1);
by (asm_full_simp_tac
(simpset() addsimps [drop_set_lift_set_Int2,
@@ -478,7 +230,7 @@
\ i: I; finite I |] \
\ ==> F : A Co B";
by (full_simp_tac (simpset() addsimps [Constrains_eq_constrains]) 1);
-by (dtac PLam_constrains_projection 1);
+by (dtac PLam_constrains_drop_set 1);
by (assume_tac 1);
by (asm_full_simp_tac
(simpset() addsimps [drop_set_INT,
@@ -498,31 +250,18 @@
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 ***)
-Goal "drop_prog i ((lift_prog i F) Join G) = F Join (drop_prog i G)";
-by (rtac program_equalityI 1);
-by (simp_tac (simpset() addsimps [Acts_Join, image_Un,
- image_compose RS sym, o_def]) 2);
-by (simp_tac (simpset() addsimps [drop_set_lift_set_Int]) 1);
-qed "drop_prog_lift_prog_Join";
-
-Goal "(lift_prog i F) Join G = lift_prog i H \
-\ ==> H = F Join (drop_prog i G)";
-by (dres_inst_tac [("f", "drop_prog i")] arg_cong 1);
-by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
-by (etac sym 1);
-qed "lift_prog_Join_eq_lift_prog_D";
-
-Goal "F : X guar Y \
-\ ==> lift_prog i F : (lift_prog i `` X) guar (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";
-
Goalw [PLam_def]
"[| ALL i:I. F i : X guar Y |] \
\ ==> (PLam I F) : (INT i: I. lift_prog i `` X) \