src/HOL/UNITY/Alloc.ML

--- a/src/HOL/UNITY/Alloc.ML	Thu Jun 17 10:41:39 1999 +0200
+++ b/src/HOL/UNITY/Alloc.ML	Thu Jun 17 10:43:05 1999 +0200
@@ -6,6 +6,7 @@
 Specification of Chandy and Charpentier's Allocator
 *)
 
+
 Addsimps [sub_def];
 
 Goalw [sysOfAlloc_def] "inj sysOfAlloc";
@@ -34,12 +35,12 @@
 AddIffs [finite_lessThan];
 
 (*could move to PPROD.ML, but function "sub" is needed there*)
-Goal "lift_set i {s. P (f s)} = {s. P (f (sub i s))}";
+Goal "lift_set i {s. P s} = {s. P (sub i s)}";
 by (asm_simp_tac (simpset() addsimps [lift_set_def]) 1);
 qed "lift_set_sub";
 
 (*ONE OF THESE IS REDUNDANT!*)
-Goal "lift_set i {s. P (f s)} = {s. P (f (s i))}";
+Goal "lift_set i {s. P s} = {s. P (s i)}";
 by (asm_simp_tac (simpset() addsimps [lift_set_def]) 1);
 qed "lift_set_sub2";
 
@@ -51,30 +52,560 @@
 by (asm_full_simp_tac (simpset() addsimps [PFUN_Stable]) 1);
 qed "PFUN_Increasing";
 
-(*FUN.ML*)
-(*left-to-right inclusion holds even if I is empty*)
-Goalw [inj_on_def]
-   "[| inj_on f C;  ALL i:I. B i <= C;  j:I |] \
-\   ==> f `` (INT i:I. B i) = (INT i:I. f `` B i)";
+
+(*The other direction fails: having FF : Stable {s. z <= f (s i)} does not
+  ensure that F has the form lift_prog i F for some F.*)
+Goal "lift_prog i `` Stable {s. P (f s)} <= Stable {s. P (f (s i))}";
+by Auto_tac;
+by (stac (lift_set_sub2 RS sym) 1); 
+by (asm_simp_tac (simpset() addsimps [lift_prog_Stable_eq]) 1);
+qed "image_lift_prog_Stable";
+
+Goal "lift_prog i `` Increasing f <= Increasing (f o sub i)";
+by (simp_tac (simpset() addsimps [Increasing_def,
+				  inj_lift_prog RS image_INT]) 1);
+by (blast_tac (claset() addIs [impOfSubs image_lift_prog_Stable]) 1);
+qed "image_lift_prog_Increasing";
+
+
+(*****************PPROD.ML******************)
+
+(*????????????????????????????????????????????????????????????????????????*)
+Goal "[| F : (drop_prog i `` X) guar (drop_prog i `` Y);  modular i Y |] \
+\     ==> lift_prog i F : X guar Y";
+by (rtac guaranteesI 1);
+by (dtac guaranteesD 1);
+by (rtac image_eqI 1);
+by (rtac (drop_prog_lift_prog_Join RS sym) 1);
+by (assume_tac 1);
+by Auto_tac;
+by (full_simp_tac (simpset() addsimps [modular_def,
+				       drop_prog_lift_prog_Join RS sym]) 1);
+by (dtac sym 1);
+by (Blast_tac 1);
+qed "drop_prog_guarantees";
+
+Goalw [constrains_def]
+     "(drop_prog i F : A co B)  =  \
+\     (F : (lift_set i A) co (lift_set i B))";
+by Auto_tac;
+by (force_tac (claset(), 
+	       simpset() addsimps [drop_act_def]) 2);
+by (blast_tac (claset() addIs [drop_act_I]) 1);
+qed "drop_prog_constrains_eq";
+
+Goal "(drop_prog i F : stable A)  =  (F : stable (lift_set i A))";
+by (simp_tac (simpset() addsimps [stable_def, drop_prog_constrains_eq]) 1);
+qed "drop_prog_stable_eq";
+
+Goal "modular i (stable {s. P(s i)})";
+by (simp_tac (simpset() addsimps [modular_def, lift_set_sub2 RS sym,
+				  drop_prog_stable_eq RS sym]) 1);
+by Auto_tac;
+qed "modular_stable_i";
+
+
+(** Weak Constrains and Stable **)
+
+Goal "f : reachable F ==> f i : reachable (drop_prog i F)";
+by (etac reachable.induct 1);
+by (force_tac (claset() addIs [reachable.Acts, drop_act_I],
+	       simpset()) 2);
+by (force_tac (claset() addIs [reachable.Init, drop_set_I],
+	       simpset()) 1);
+qed "reachable_imp_reachable_drop_prog";
+
+Goalw [Constrains_def]
+     "drop_prog i F : A Co B ==> F : (lift_set i A) Co (lift_set i B)";
+by (full_simp_tac (simpset() addsimps [drop_prog_constrains_eq]) 1);
+by (etac constrains_weaken_L 1);
+by Auto_tac;
+by (etac reachable_imp_reachable_drop_prog 1);
+qed "drop_prog_Constrains_D";
+
+Goalw [Stable_def]
+     "drop_prog i F : Stable A ==> F : Stable (lift_set i A)";
+by (asm_simp_tac (simpset() addsimps [drop_prog_Constrains_D]) 1);
+qed "drop_prog_Stable_D";
+
+(***????????????????
+Goal "i ~= j ==> lift_prog i `` UNIV <= stable {s. P (f (s j))}";
+by Auto_tac;
+by (stac (lift_set_sub2 RS sym) 1); 
+by (asm_simp_tac (simpset() addsimps [lift_prog_Stable_eq]) 1);
+qed "image_lift_prog_Stable";
+????????????????*)
+
+
+
+(****UNITY.ML?****)
+Goalw [constrains_def, Idle_def] "F : Idle ==> (F : A co B) = (A<=B)";
+by (Blast_tac 1);
+qed "constrains_Idle_iff";
+
+Goalw [stable_def,constrains_def,Idle_def] "F : Idle ==> F : stable A";
+by (Blast_tac 1);
+qed "stable_Idle";
+
+Goalw [drop_act_def, lift_act_def]
+     "i~=j ==> drop_act i (lift_act j act) <= Id";
+by Auto_tac;
+qed "neq_drop_act_lift_act";
+
+
+Goal "drop_prog i (JN j:I - {i}. lift_prog j F) : Idle";
+by (simp_tac (simpset() addsimps [Acts_JN,Idle_def]) 1);
+by Auto_tac;
+by (rtac ccontr 1);  
+  (*otherwise PROOF FAILED because tactics don't get negated conclusion*)
+by (blast_tac (claset() addSDs [impOfSubs neq_drop_act_lift_act]) 1);
+qed "drop_prog_JN_other_Idle";
+
+Goal "[| F component G;  G : Idle |] ==> F : Idle";
+by (full_simp_tac (simpset() addsimps [Idle_def, component_eq_subset]) 1);
+by Auto_tac;
+qed "component_Idle";
+
+Goal "G component (JN j: I-{i}. JN H: UNIV. lift_prog j H) \
+\     ==> drop_prog i G : Idle";
+by (dtac drop_prog_mono 1);
+be component_Idle 1;
+by (simp_tac (simpset() addsimps [drop_prog_JN_other_Idle, 
+				  lift_prog_JN RS sym]) 1);
+qed "component_JN_neq_imp_Idle";
+
+Goal "drop_prog i G : Idle ==> G : stable (lift_set i A)";
+by (simp_tac (simpset() addsimps [drop_prog_stable_eq RS sym]) 1);
+by (blast_tac (claset() addIs [stable_Idle]) 1);
+qed "Idle_imp_stable_lift_set";
+
+xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx;
+
+Goalw [guarantees_def]
+    "[| F : (X Int A) guar (Y Int A);  \
+\       F : (X Int B) guar (Y Int B);  \
+\       UNIV <= A Un B |] \
+\    ==> F : X guar Y";
 by (Blast_tac 1);
-qed "inj_on_image_INT";
+qed "guarantees_Un_partition_I";
+
+Goalw [guarantees_def]
+    "[| ALL i:I. F : (X Int A i) guar (Y Int A i);  UNIV <= UNION I A |] \
+\    ==> F : X guar Y";
+by (Blast_tac 1);
+qed "guarantees_UN_partition_I";
+
+Goal "{G. G component (JN j: I-{i}. JN H: UNIV. lift_prog j H)} \
+\     <= stable (lift_set i A)";
+by (Clarify_tac 1);
+by (etac (component_JN_neq_imp_Idle RS Idle_imp_stable_lift_set) 1);
+result();
+
+Goal "i~=j ==> (lift_prog j `` UNIV) <= Increasing (f o sub i)";
+by (rewtac Increasing_def);
+by Auto_tac;
+by (stac (lift_set_sub2 RS sym) 1); 
+by (blast_tac (claset() addIs [neq_imp_lift_prog_Stable]) 1);
+result();
+
+Goal "Cl : UNIV guar Increasing f   ==> \
+\ lift_prog i Cl : {G. G component (JN j: I. JN H: UNIV. lift_prog j H)} \
+\                  guar Increasing (f o sub i)";
+by (rtac guarantees_partition_I 1);
+by (simp_tac (simpset() addsimps []) 1);
+
+by (dtac lift_prog_guarantees 1);
+by (etac guarantees_weaken 1);
+by Auto_tac;
+by (rtac (impOfSubs image_lift_prog_Increasing) 1);
+by Auto_tac;
+result();
+
+
+
+
+
+
+????????????????????????????????????????????????????????????????;
+
+
+
+
+Goal "drop_prog i `` UNIV = UNIV";
+by Auto_tac;
+fr image_eqI;
+by (rtac (lift_prog_inverse RS sym) 1);
+by Auto_tac;
+result();
+
+
+Goal "[| F : (drop_prog i `` XX) guar Y;    \
+\        ALL Z:XX. Z component (JN i:UNIV. JN G:UNIV. lift_prog i G) |] \
+\     ==> lift_prog i F : XX guar (lift_prog i `` Y)";
+by (rtac guaranteesI 1);
+by (dtac guaranteesD 1);
+by (ball_tac 1);
+by (full_simp_tac (simpset() addsimps [Join_component_iff]) 1);
+by (full_simp_tac (simpset() addsimps [component_eq_subset, Acts_JN]) 1);
+
+by (rtac image_eqI 2);
+by (assume_tac 3);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join RS sym]) 2);
+
+by (rtac (drop_prog_lift_prog_Join RS sym) 1);
+by (assume_tac 1);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join RS sym]) 1);
+by (rtac image_eqI 1);
+by (assume_tac 2);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+
+by (rtac guaranteesI 1);
+by (dtac guaranteesD 1);
+by (rtac image_eqI 1);
+by (rtac (drop_prog_lift_prog_Join RS sym) 1);
+by (assume_tac 1);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join RS sym]) 1);
+by (rtac image_eqI 1);
+by (assume_tac 2);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+
+
+by (dtac sym 1);
+by (Blast_tac 1);
+
+
+by (rtac guaranteesI 1);
+by (dtac guaranteesD 1);
+by (rtac image_eqI 2);
+by (assume_tac 3);
+
+
+Goal "F : X guar Y \
+\     ==> lift_prog i F : XX guar (lift_prog i `` Y)";
+by (rtac guaranteesI 1);
+by (dtac guaranteesD 1);
+by (rtac image_eqI 2);
+by (assume_tac 3);
+
+by Auto_tac;
+by (blast_tac (claset() addDs [lift_prog_Join_eq_lift_prog_D, guaranteesD]) 1);
+qed "lift_prog_guarantees";
+
+
+
+
+
+Goal "x : reachable (drop_prog i F) ==> x : drop_set i (reachable F)";
+by (etac reachable.induct 1);
+by Auto_tac;
+by (rewtac drop_set_def);
+by (force_tac (claset() addIs [reachable.Init, drop_set_I],
+	       simpset()) 1);
+by (rewtac drop_act_def);
+by Auto_tac;
+by (rtac imageI 1);
+by (rtac reachable.Acts 1);
+by (assume_tac 1);
+by (assume_tac 1);
+
+
+Goal "reachable (drop_prog i F) = drop_set i (reachable F)";
+by (rewtac drop_set_def);
+by Auto_tac;
+by (etac reachable_imp_reachable_drop_prog 2);
+
+
+(*FAILS, so the X condition is too strong*)
+Goal "((f o sub i) localTo (lift_prog i F))  <= (UN F. {lift_prog i F})";
+by (simp_tac (simpset() addsimps [localTo_def]) 1);
+by Auto_tac;
+
+
+Goal "lift_prog i `` drop_prog i `` ((f o sub i) localTo (lift_prog i F)) \
+\     <= ((f o sub i) localTo (lift_prog i F))";
+by (simp_tac (simpset() addsimps [localTo_def]) 1);
+by Auto_tac;
 
 
 
-Goal "lift_prog i `` Increasing f <= Increasing (f o sub i)";
-by (asm_simp_tac 
-    (simpset() addsimps [Increasing_def,
-			 inj_lift_prog RS inj_on_image_INT]) 1);
-auto();
-by (dres_inst_tac [("x","z")] spec 1);
-auto();
-by (asm_simp_tac (simpset() addsimps [lift_set_sub2 RS sym,
+Goalw [Constrains_def]
+     "(F : (lift_set i A) Co (lift_set i B)) \
+\     ==> (drop_prog i F : A Co B)";
+by (full_simp_tac (simpset() addsimps [drop_prog_constrains_eq]) 1);
+by (etac constrains_weaken_L 1);
+
+by Auto_tac;
+by (force_tac (claset(), 
+	       simpset() addsimps [drop_act_def]) 2);
+by (blast_tac (claset() addIs [drop_act_I]) 1);
+qed "drop_prog_Constrains_eq";
+
+
+(*Does NOT appear to be provable, so this notion of modular is too strong*)
+Goal "lift_prog i `` drop_prog i `` Increasing (f o sub i) \
+\        <= Increasing (f o sub i)";
+by (simp_tac (simpset() addsimps [Increasing_def]) 1);
+by Auto_tac;
+by (stac (lift_set_sub2 RS sym) 1); 
+by (simp_tac (simpset() addsimps [lift_prog_Stable_eq]) 1);
+
+
+
+
+
+Goal "[| F : (drop_prog i `` X) guar (drop_prog i `` Y);  \
+\        X <= (UN F. {lift_prog i F});  \
+\        lift_prog i `` drop_prog i `` Y <= Y |] \
+\     ==> lift_prog i F : X guar Y";
+by (rtac guaranteesI 1);
+by (set_mp_tac 1);
+by (Clarify_tac 1);
+by (dtac guaranteesD 1);
+by (rtac image_eqI 1);
+by (rtac (drop_prog_lift_prog_Join RS sym) 1);
+by (assume_tac 1);
+by Auto_tac;
+by (asm_full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join RS sym]) 1);
+by Auto_tac;
+qed "drop_prog_guarantees";
+
+
+
+NEW NOTION OF MODULAR
+
+
+Goal "(ALL H:X. lift_prog i (drop_prog i H) : X) = \
+\     (lift_prog i `` drop_prog i `` X <= X)";
+by Auto_tac;
+qed "modular_equiv";
+
+Goal "lift_prog i `` drop_prog i `` (lift_prog i `` F) <= lift_prog i `` F";
+by (simp_tac (simpset() addsimps [image_compose RS sym, o_def]) 1);
+result();
+
+Goal "[| F : (drop_prog i `` X) guar (drop_prog i `` Y);  \
+\        lift_prog i `` drop_prog i `` X <= X;  \
+\        lift_prog i `` drop_prog i `` Y <= Y|] \
+\     ==> lift_prog i F : X guar Y";
+by (full_simp_tac (simpset() addsimps [modular_equiv RS sym]) 1);
+by (rtac guaranteesI 1);
+by (ball_tac 1);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+by (dtac guaranteesD 1);
+by (rtac image_eqI 1);
+by (assume_tac 2);back();
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+by (ALLGOALS Clarify_tac);
+by (dres_inst_tac [("f", "lift_prog i")] arg_cong 1);
+
+by (rtac guaranteesI 1);
+by (dtac guaranteesD 1);
+by (ALLGOALS Clarify_tac);
+by (rtac image_eqI 1);
+by (rtac (drop_prog_lift_prog_Join RS sym) 1);
+by (assume_tac 1);
+by (full_simp_tac (simpset() addsimps [modular_equiv RS sym]) 1);
+by (REPEAT (ball_tac 1));
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+ren "H" 1;
+by (dtac sym 1);
+by (asm_full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+????????????????;
+
+by Auto_tac;
+
+by (dres_inst_tac [("f", "drop_prog i")] arg_cong 1);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+by (assume_tac 1);
+by (ALLGOALS Clarify_tac);
+by (set_mp_tac 1);
+by (ALLGOALS Clarify_tac);
+by Auto_tac;
+by (dres_inst_tac [("f", "drop_prog i")] arg_cong 1);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+by Auto_tac;
+qed "drop_prog_guarantees";
+
+
+
+
+
+
+
+
+Goal "(lift_prog i `` F) <= lift_prog i `` drop_prog i `` (lift_prog i `` F)";
+by (simp_tac (simpset() addsimps [image_compose RS sym, o_def]) 1);
+by Auto_tac;
+result();
+
+
+Goal "H : lift_prog i `` drop_prog i `` X";
+
+
+Goal "[| F : (drop_prog i `` X) guar (drop_prog i `` Y);  \
+\        X <= lift_prog i `` drop_prog i `` X;  \
+\        Y <= lift_prog i `` drop_prog i `` Y |] \
+\     ==> lift_prog i F : X guar Y";
+by (rtac guaranteesI 1);
+by (set_mp_tac 1);
+by (ALLGOALS Clarify_tac);
+by (dtac guaranteesD 1);
+by (rtac image_eqI 1);
+by (dres_inst_tac [("f", "drop_prog i")] arg_cong 1);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+by (assume_tac 1);
+by (ALLGOALS Clarify_tac);
+by (set_mp_tac 1);
+by (ALLGOALS Clarify_tac);
+by Auto_tac;
+by (dres_inst_tac [("f", "drop_prog i")] arg_cong 1);
+by (full_simp_tac (simpset() addsimps [drop_prog_lift_prog_Join]) 1);
+by Auto_tac;
+qed "drop_prog_guarantees";
+
+
+
+(*FAILS*)
+Goal "modular i (lift_prog i `` Y)";
+
+
+
+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!;
+(*Precondition is TOO STRONG*)
+Goal "Cl : UNIV guar Increasing f \
+\     ==>lift_prog i Cl : (lift_prog i `` UNIV) guar Increasing (f o sub i)";
+by (rtac drop_prog_guarantees 1);
+by (etac guarantees_weaken 1);
+by Auto_tac;
+by (rtac (impOfSubs image_lift_prog_Increasing) 2);
+by (rtac image_eqI 1);
+by (rtac (lift_prog_inverse RS sym) 1);
+by (rtac (impOfSubs image_lift_prog_Increasing) 1);
+by Auto_tac;
+fr imageI;
+
+by (stac (lift_set_sub2 RS sym) 1); 
+by (asm_simp_tac (simpset() addsimps [lift_prog_Stable_eq]) 1);
+
+
+
+
+
+
+Goal "Cl : UNIV guar Increasing f \
+\     ==>lift_prog i Cl : UNIV guar Increasing (f o sub i)";
+by (rtac drop_prog_guarantees 1);
+by (rewtac Increasing_def);
+by (etac guarantees_weaken 1);
+by Auto_tac;
+by (rtac image_eqI 1);
+by (rtac (lift_prog_inverse RS sym) 1);
+by Auto_tac;
+by (stac (lift_set_sub2 RS sym) 1); 
+by (asm_simp_tac (simpset() addsimps [lift_prog_Stable_eq]) 1);
+
+
+
+(*could move to PPROD.ML, but function "sub" is needed there*)
+Goalw [drop_set_def] "drop_set i {s. P (f (sub i s))} = {s. P (f s)}";
+by Auto_tac;
+qed "drop_set_sub";
+
+Goal "lift_set i (reachable (drop_prog i F)) = reachable F";
+
+(*False?*)
+Goal "modular i ((f o sub i) localTo (lift_prog i F))";
+by (simp_tac (simpset() addsimps [localTo_def, modular_def]) 1);
+by Auto_tac;
+
+
+
+(*FAILS*)
+Goal "(drop_prog i F : A Co B)  =  \
+\     (F : (lift_set i A) Co (lift_set i B))";
+by (simp_tac (simpset() addsimps [Constrains_def, drop_prog_constrains_eq,
+				  lift_set_Int]) 1);
+
+??
+by (simp_tac (simpset() addsimps [Constrains_def, reachable_drop_prog,
+				  drop_set_Int RS sym,
+				  drop_prog_constrains_eq]) 1);
+qed "drop_prog_Constrains_eq";
+
+(*FAILS*)
+Goal "(drop_prog i F : Stable A)  =  (F : Stable (lift_set i A))";
+by (simp_tac (simpset() addsimps [Stable_def, drop_prog_Constrains_eq]) 1);
+qed "drop_prog_Stable_eq";
+
+
+(*FAILS*)
+Goal "modular i (Stable {s. P(s i)})";
+by (full_simp_tac (simpset() addsimps [modular_def,
+				       drop_prog_lift_prog_Join RS sym]) 1);
+by (full_simp_tac (simpset() addsimps [lift_set_sub2 RS sym,
 				      lift_prog_Stable_eq]) 1);
-qed "image_lift_prog_Increasing";
+by Auto_tac;
+
+
+
+Goal "modular i (Increasing (rel o sub i))";
+by (full_simp_tac (simpset() addsimps [modular_def, Increasing_def,
+				       drop_prog_lift_prog_Join RS sym]) 1);
+
+
+Goal "modular i (lift_prog i `` X)";
+by (full_simp_tac (simpset() addsimps [modular_def,
+				       drop_prog_lift_prog_Join RS sym]) 1);
+
 
 
 Goal "i < Nclients ==> \
-\     lift_prog i Client : range (lift_prog i) guar Increasing (rel o sub i)";
+\     lift_prog i Client : UNIV guar Increasing (rel o sub i)";
+
+
+
+
+
+
+Goal "(f o sub i) localTo (lift_prog i F) <= lift_prog i `` (f localTo F)";
+
+
+
+
+
+
+Goal "(f o sub i) localTo (lift_prog i F) <= lift_prog i `` (f localTo F)";
+by (simp_tac (simpset() addsimps [localTo_def]) 1);
+by Auto_tac;
+by (dres_inst_tac [("x","z")] spec 1);
+by Auto_tac;
+by (asm_simp_tac (simpset() addsimps [lift_set_sub2 RS sym,
+				      lift_prog_stable_eq]) 1);
+qed "image_lift_prog_?";
+
+Goal "{GX. EX G: UNIV. lift_prog i G component GX} = UNIV";
+by Auto_tac;
+by (asm_full_simp_tac (simpset() addsimps [component_eq_subset]) 1);
+by (asm_full_simp_tac (simpset() addsimps [lift_set_def]) 1);
+ren "GX" 1;
+by (res_inst_tac [("x", "mk_program((%f. f i)``(Init GX), drop_act i `` Acts GX)")] exI 1);
+by Auto_tac;
+
+
+by (stac Acts_eq 1);
+
+fr conjI;
+by (Clarify_tac 1);
+by (stac Init_eq 1);
+
+
+{GX. EX G:X. lift_prog i G component GX}
+{GY. EX G:Y. lift_prog i G component GY}
+
+
+
+uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu;
+