# HG changeset patch
# User paulson
# Date 930127175 -7200
# Node ID 56e08e2649617e1d7c78a1f77ecc98424897f412
# Parent  5a557122bb62eb46b9dd333d6c12a4979bf4ede0
renamed PPI to plam
simplified the definition of lift_act

diff -r 5a557122bb62 -r 56e08e264961 src/HOL/UNITY/PPROD.ML
--- a/src/HOL/UNITY/PPROD.ML	Wed Jun 23 10:38:49 1999 +0200
+++ b/src/HOL/UNITY/PPROD.ML	Wed Jun 23 10:39:35 1999 +0200
@@ -31,6 +31,9 @@
 
 Goalw [lift_act_def] "lift_act i Id = Id";
 by Auto_tac;
+be rev_mp 1;
+bd sym 1;
+by (asm_simp_tac (simpset() addsimps [fun_upd_idem_iff]) 1);
 qed "lift_act_Id";
 Addsimps [lift_act_Id];
 
@@ -110,10 +113,32 @@
 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 (REPEAT_FIRST (resolve_tac [exI, conjI]));
-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);
+br ext 1;
+by (Asm_simp_tac 1);
 qed "lift_act_inverse";
 Addsimps [lift_act_inverse];
 
@@ -136,43 +161,47 @@
 qed "inj_lift_prog";
 
 
-(** PPROD **)
+(** PLam **)
 
-Goalw [PPROD_def] "Init (PPROD I F) = (INT i:I. lift_set i (Init (F i)))";
+Goalw [PLam_def] "Init (PLam I F) = (INT i:I. lift_set i (Init (F i)))";
 by Auto_tac;
-qed "Init_PPROD";
-Addsimps [Init_PPROD];
+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 (Blast_tac 1);
+by Auto_tac;
+br exI 1;
+by Auto_tac;
 qed "lift_act_eq";
 AddIffs [lift_act_eq];
 
-Goal "Acts (PPROD I F) = insert Id (UN i:I. lift_act i `` Acts (F i))";
+
+Goal "Acts (PLam I F) = insert Id (UN i:I. lift_act i `` Acts (F i))";
 by (auto_tac (claset(),
-	      simpset() addsimps [PPROD_def, Acts_JN]));
-qed "Acts_PPROD";
+	      simpset() addsimps [PLam_def, Acts_JN]));
+qed "Acts_PLam";
 
-Goal "PPROD {} F = SKIP";
-by (simp_tac (simpset() addsimps [PPROD_def]) 1);
-qed "PPROD_empty";
+Goal "PLam {} F = SKIP";
+by (simp_tac (simpset() addsimps [PLam_def]) 1);
+qed "PLam_empty";
 
-Goal "(PPI i: I. SKIP) = SKIP";
-by (simp_tac (simpset() addsimps [PPROD_def,lift_prog_SKIP,JN_constant]) 1);
-qed "PPROD_SKIP";
+Goal "(plam i: I. SKIP) = SKIP";
+by (simp_tac (simpset() addsimps [PLam_def,lift_prog_SKIP,JN_constant]) 1);
+qed "PLam_SKIP";
 
-Addsimps [PPROD_SKIP, PPROD_empty];
+Addsimps [PLam_SKIP, PLam_empty];
 
-Goalw [PPROD_def]
-    "PPROD (insert i I) F = (lift_prog i (F i)) Join (PPROD I F)";
+Goalw [PLam_def]
+    "PLam (insert i I) F = (lift_prog i (F i)) Join (PLam I F)";
 by Auto_tac;
-qed "PPROD_insert";
+qed "PLam_insert";
 
-Goalw [PPROD_def] "i : I ==> (lift_prog i (F i)) component (PPROD I F)";
+Goalw [PLam_def] "i : I ==> (lift_prog i (F i)) component (PLam I F)";
 (*blast_tac doesn't use HO unification*)
 by (fast_tac (claset() addIs [component_JN]) 1);
-qed "component_PPROD";
+qed "component_PLam";
 
 
 (*** Safety: co, stable, invariant ***)
@@ -219,19 +248,19 @@
 	  neq_imp_lift_prog_stable RS stable_imp_Stable);
 
 
-(** Safety and PPROD **)
+(** Safety and PLam **)
 
 Goal "i : I ==>  \
-\     (PPROD I F : (lift_set i A) co (lift_set i B))  =  \
+\     (PLam I F : (lift_set i A) co (lift_set i B))  =  \
 \     (F i : A co B)";
-by (asm_simp_tac (simpset() addsimps [PPROD_def, constrains_JN]) 1);
+by (asm_simp_tac (simpset() addsimps [PLam_def, constrains_JN]) 1);
 by (blast_tac (claset() addIs [lift_prog_constrains_eq RS iffD1, 
 			       constrains_imp_lift_prog_constrains]) 1);
-qed "PPROD_constrains";
+qed "PLam_constrains";
 
-Goal "i : I ==> (PPROD I F : stable (lift_set i A)) = (F i : stable A)";
-by (asm_simp_tac (simpset() addsimps [stable_def, PPROD_constrains]) 1);
-qed "PPROD_stable";
+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";
 
 
 (** Safety, lift_prog + drop_set **)
@@ -244,51 +273,50 @@
 	       simpset()) 1);
 qed "lift_prog_constrains_projection";
 
-Goal "[| PPROD I F : AA co BB;  i: I |] \
+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);
 (*rotate this assumption to be last*)
-by (dres_inst_tac [("psi", "PPROD I F : ?C")] asm_rl 1);
-by (asm_full_simp_tac (simpset() addsimps [PPROD_def, constrains_JN]) 1);
-qed "PPROD_constrains_projection";
+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";
 
 
 (** invariant **)
 
-(*UNUSED*)
 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 |] \
-\     ==> PPROD I F : invariant (lift_set i A)";
+\     ==> PLam I F : invariant (lift_set i A)";
 by (auto_tac (claset(),
-	      simpset() addsimps [invariant_def, PPROD_stable]));
-qed "invariant_imp_PPROD_invariant";
+	      simpset() addsimps [invariant_def, PLam_stable]));
+qed "invariant_imp_PLam_invariant";
 
 (*The f0 premise ensures that the product is well-defined.*)
-Goal "[| PPROD I F : invariant (lift_set i A);  i : I;  \
-\        f0: Init (PPROD I F) |] ==> F i : invariant A";
+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, PPROD_stable]));
+	      simpset() addsimps [invariant_def, PLam_stable]));
 by (dres_inst_tac [("c", "f0(i:=x)")] subsetD 1);
 by Auto_tac;
-qed "PPROD_invariant_imp_invariant";
+qed "PLam_invariant_imp_invariant";
 
-Goal "[| i : I;  f0: Init (PPROD I F) |] \
-\     ==> (PPROD I F : invariant (lift_set i A)) = (F i : invariant A)";
-by (blast_tac (claset() addIs [invariant_imp_PPROD_invariant, 
-			       PPROD_invariant_imp_invariant]) 1);
-qed "PPROD_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 \
-\     ==> ((PPI x:I. F) : invariant (lift_set i A)) = (F : invariant A)";
+\     ==> ((plam x:I. F) : invariant (lift_set i A)) = (F : invariant A)";
 by (auto_tac (claset(),
-	      simpset() addsimps [invariant_def, PPROD_stable]));
-qed "PFUN_invariant";
+	      simpset() addsimps [invariant_def, PLam_stable]));
+qed "const_PLam_invariant";
 
 
 (*** Substitution Axiom versions: Co, Stable ***)
@@ -330,163 +358,150 @@
 qed "lift_prog_Stable_eq";
 
 
-(** Reachability for PPROD **)
+(** Reachability for PLam **)
 
-Goal "[| f : reachable (PPROD I F);  i : I |] ==> f i : reachable (F i)";
+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() addsimps [Acts_PPROD]));
-qed "reachable_PPROD";
+     simpset() addsimps [Acts_PLam]));
+qed "reachable_PLam";
 
-Goal "reachable (PPROD I F) <= {f. ALL i:I. f i : reachable (F i)}";
-by (force_tac (claset() addSDs [reachable_PPROD], simpset()) 1);
-qed "reachable_PPROD_subset1";
+(*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))";
+auto();
+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 (PPROD I F)      \
-\             --> f(i:=A) : reachable (lift_prog i (F i) Join PPROD I F)";
+\  ==> 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 PPROD F case*)
+(*Init of F, action of PLam F case*)
 by (rtac reachable.Acts 1);
 by (force_tac (claset(), simpset() addsimps [Acts_Join]) 1);
 by (assume_tac 1);
-by (force_tac (claset() addIs [ext], simpset() addsimps [Acts_PPROD]) 1);
+by (force_tac (claset() addIs [ext], simpset() addsimps [Acts_PLam]) 1);
 (*induction over the 2nd "reachable" assumption*)
 by (eres_inst_tac [("xa","f")] reachable.induct 1);
-(*Init of PPROD F, action of F case*)
+(*Init of PLam F, action of F case*)
 by (res_inst_tac [("act","lift_act i act")] reachable.Acts 1);
 by (force_tac (claset(), simpset() addsimps [Acts_Join]) 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 PPROD I F*)
+(*last case: an action of PLam I F*)
 by (rtac reachable.Acts 1);
 by (force_tac (claset(), simpset() addsimps [Acts_Join]) 1);
 by (assume_tac 1);
-by (force_tac (claset() addIs [ext], simpset() addsimps [Acts_PPROD]) 1);
-qed_spec_mp "reachable_lift_Join_PPROD";
+by (force_tac (claset() addIs [ext], simpset() addsimps [Acts_PLam]) 1);
+qed_spec_mp "reachable_lift_Join_PLam";
 
 
 (*The index set must be finite: otherwise infinitely many copies of F can
-  perform actions, and PPROD can never catch up in finite time.*)
+  perform actions, and PLam can never catch up in finite time.*)
 Goal "finite I \
-\     ==> {f. ALL i:I. f i : reachable (F i)} <= reachable (PPROD I F)";
+\     ==> (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_PPROD], 
-	       simpset() addsimps [PPROD_insert]) 1);
-qed "reachable_PPROD_subset2";
+by (force_tac (claset() addDs [reachable_lift_Join_PLam], 
+	       simpset() addsimps [PLam_insert]) 1);
+qed "reachable_PLam_subset2";
 
 Goal "finite I ==> \
-\     reachable (PPROD I F) = {f. ALL i:I. f i : reachable (F i)}";
+\     reachable (PLam I F) = (INT i:I. lift_set i (reachable (F i)))";
 by (REPEAT_FIRST (ares_tac [equalityI,
-			    reachable_PPROD_subset1, 
-			    reachable_PPROD_subset2]));
-qed "reachable_PPROD_eq";
+			    reachable_PLam_subset1, 
+			    reachable_PLam_subset2]));
+qed "reachable_PLam_eq";
 
 
 (** Co **)
 
 Goal "[| F i : A Co B;  i: I;  finite I |]  \
-\     ==> PPROD I F : (lift_set i A) Co (lift_set i B)";
+\     ==> 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_PPROD_eq]));
+			 reachable_PLam_eq]));
 by (auto_tac (claset(), 
-              simpset() addsimps [constrains_def, PPROD_def, Acts_JN]));
+              simpset() addsimps [constrains_def, PLam_def, Acts_JN]));
 by (REPEAT (blast_tac (claset() addIs reachable.intrs) 1));
-qed "Constrains_imp_PPROD_Constrains";
+qed "Constrains_imp_PLam_Constrains";
 
-Goal "[| ALL i:I. f0 i : R i;   i: I |] \
-\     ==> drop_set i ({f. (ALL i:I. f i : R i)} Int lift_set i A) = R i Int A";
+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 [rinst [("x", "?ff(i := ?u)")] image_eqI],
-	       simpset() addsimps [drop_set_def, lift_set_def]) 1);
-qed "drop_set_Int_depend";
+	       simpset() addsimps [drop_set_def]) 1);
+qed "drop_set_INT_lift_set";
 
-(*Again, we need the f0 premise so that PPROD I F has an initial state;
+(*Again, we need the f0 premise so that PLam I F has an initial state;
   otherwise its Co-property is vacuous.*)
-Goal "[| PPROD I F : (lift_set i A) Co (lift_set i B);  \
-\        i: I;  finite I;  f0: Init (PPROD I F) |]  \
+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 PPROD_constrains_projection 1);
+by (dtac PLam_constrains_projection 1);
 by (assume_tac 1);
 by (asm_full_simp_tac
-    (simpset() addsimps [drop_set_Int_depend, reachable_PPROD_eq]) 1);
-qed "PPROD_Constrains_imp_Constrains";
+    (simpset() addsimps [drop_set_lift_set_Int2,
+			 drop_set_INT_lift_set, reachable_PLam_eq]) 1);
+qed "PLam_Constrains_imp_Constrains";
 
 
-Goal "[| i: I;  finite I;  f0: Init (PPROD I F) |]  \
-\     ==> (PPROD I F : (lift_set i A) Co (lift_set i B)) =  \
+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_PPROD_Constrains, 
-			       PPROD_Constrains_imp_Constrains]) 1);
-qed "PPROD_Constrains";
+by (blast_tac (claset() addIs [Constrains_imp_PLam_Constrains, 
+			       PLam_Constrains_imp_Constrains]) 1);
+qed "PLam_Constrains";
 
-Goal "[| i: I;  finite I;  f0: Init (PPROD I F) |]  \
-\     ==> (PPROD I F : Stable (lift_set i A)) = (F i : Stable A)";
-by (asm_simp_tac (simpset() delsimps [Init_PPROD]
-			    addsimps [Stable_def, PPROD_Constrains]) 1);
-qed "PPROD_Stable";
+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";
 
 
-(** PFUN (no dependence on i) doesn't require the f0 premise **)
+(** const_PLam (no dependence on i) doesn't require the f0 premise **)
 
-Goal "i: I \
-\     ==> drop_set i ({f. (ALL i:I. f i : R)} Int lift_set i A) = R Int A";
-by (force_tac (claset(), simpset() addsimps [drop_set_def]) 1);
-qed "drop_set_Int";
-
-Goal "[| (PPI x:I. F) : (lift_set i A) Co (lift_set i B);  \
+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 PPROD_constrains_projection 1);
+by (dtac PLam_constrains_projection 1);
 by (assume_tac 1);
 by (asm_full_simp_tac
-    (simpset() addsimps [drop_set_Int, Collect_conj_eq RS sym,
-			 reachable_PPROD_eq]) 1);
-qed "PFUN_Constrains_imp_Constrains";
+    (simpset() addsimps [drop_set_INT,
+			 drop_set_lift_set_Int2, Collect_conj_eq RS sym,
+			 reachable_PLam_eq]) 1);
+qed "const_PLam_Constrains_imp_Constrains";
 
 Goal "[| i: I;  finite I |]  \
-\     ==> ((PPI x:I. F) : (lift_set i A) Co (lift_set i B)) =  \
+\     ==> ((plam x:I. F) : (lift_set i A) Co (lift_set i B)) =  \
 \         (F : A Co B)";
-by (blast_tac (claset() addIs [Constrains_imp_PPROD_Constrains, 
-			       PFUN_Constrains_imp_Constrains]) 1);
-qed "PFUN_Constrains";
+by (blast_tac (claset() addIs [Constrains_imp_PLam_Constrains, 
+			       const_PLam_Constrains_imp_Constrains]) 1);
+qed "const_PLam_Constrains";
 
 Goal "[| i: I;  finite I |]  \
-\     ==> ((PPI x:I. F) : Stable (lift_set i A)) = (F : Stable A)";
-by (asm_simp_tac (simpset() addsimps [Stable_def, PFUN_Constrains]) 1);
-qed "PFUN_Stable";
+\     ==> ((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";
 
 
 
 (*** guarantees properties ***)
 
-Goal "drop_act i (lift_act i act) = act";
-by (force_tac (claset() addSIs [rinst [("x", "?ff(i := ?u)")] exI],
-	       simpset() addsimps [drop_act_def, lift_act_def]) 1);
-qed "lift_act_inverse";
-Addsimps [lift_act_inverse];
-
-
-(*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";
-
 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,
@@ -508,16 +523,16 @@
 by (blast_tac (claset() addDs [lift_prog_Join_eq_lift_prog_D, guaranteesD]) 1);
 qed "lift_prog_guarantees";
 
-Goalw [PPROD_def]
+Goalw [PLam_def]
     "[| ALL i:I. F i : X guar Y |] \
-\    ==> (PPROD I F) : (INT i: I. lift_prog i `` X) \
+\    ==> (PLam I F) : (INT i: I. lift_prog i `` X) \
 \                 guar (INT i: I. lift_prog i `` Y)";
 by (blast_tac (claset() addIs [guarantees_JN_INT, lift_prog_guarantees]) 1);
-qed "guarantees_PPROD_INT";
+qed "guarantees_PLam_INT";
 
-Goalw [PPROD_def]
+Goalw [PLam_def]
     "[| ALL i:I. F i : X guar Y |] \
-\    ==> (PPROD I F) : (UN i: I. lift_prog i `` X) \
+\    ==> (PLam I F) : (UN i: I. lift_prog i `` X) \
 \                 guar (UN i: I. lift_prog i `` Y)";
 by (blast_tac (claset() addIs [guarantees_JN_UN, lift_prog_guarantees]) 1);
-qed "guarantees_PPROD_UN";
+qed "guarantees_PLam_UN";
diff -r 5a557122bb62 -r 56e08e264961 src/HOL/UNITY/PPROD.thy
--- a/src/HOL/UNITY/PPROD.thy	Wed Jun 23 10:38:49 1999 +0200
+++ b/src/HOL/UNITY/PPROD.thy	Wed Jun 23 10:39:35 1999 +0200
@@ -17,7 +17,7 @@
     "drop_set i A == (%f. f i) `` A"
 
   lift_act :: "['a, ('b*'b) set] => (('a=>'b) * ('a=>'b)) set"
-    "lift_act i act == {(f,f'). EX s'. f' = f(i:=s') & (f i, s') : act}"
+    "lift_act i act == {(f,f'). f(i:= f' i) = f' & (f i, f' i) : act}"
 
   drop_act :: "['a, (('a=>'b) * ('a=>'b)) set] => ('b*'b) set"
     "drop_act i act == (%(f,f'). (f i, f' i)) `` act"
@@ -33,13 +33,13 @@
 		   drop_act i `` (Acts F))"
 
   (*products of programs*)
-  PPROD  :: ['a set, 'a => 'b program] => ('a => 'b) program
-    "PPROD I F == JN i:I. lift_prog i (F i)"
+  PLam  :: ['a set, 'a => 'b program] => ('a => 'b) program
+    "PLam I F == JN i:I. lift_prog i (F i)"
 
 syntax
-  "@PPROD" :: [pttrn, 'a set, 'b set] => ('a => 'b) set ("(3PPI _:_./ _)" 10)
+  "@PLam" :: [pttrn, 'a set, 'b set] => ('a => 'b) set ("(3plam _:_./ _)" 10)
 
 translations
-  "PPI x:A. B"   == "PPROD A (%x. B)"
+  "plam x:A. B"   == "PLam A (%x. B)"
 
 end