NotationPt.Ident (id,None)
;;
+let mk_sym s = NotationPt.Symbol (s,0);;
+
let rec mk_prods l t =
match l with
[] -> t
status#set_obj(u,h,NCicUntrusted.apply_subst_metasenv status subst metasenv,subst,o)
;;
+(* needed to workaround a weakness of the refiner? *)
+let rec generalize0_tac tl s =
+ match tl with
+ | [] -> s
+ | t0::tl0 -> NTactics.generalize0_tac [t0] (generalize0_tac tl0 s)
+;;
+
+
(* input: nome della variabile riscritta
* output: lista dei nomi delle variabili il cui tipo dipende dall'input *)
let cascade_select_in_ctx status ~subst ctx skip iname =
let nargs it nleft consno =
pp (lazy (Printf.sprintf "nargs %d %d" nleft consno));
- let _,indname,_,cl = it in
+ let _,_indname,_,cl = it in
let _,_,t_k = List.nth cl consno in
List.length (arg_list nleft t_k) ;;
let default_pattern = "",0,(None,[],Some NotationPt.UserInput);;
let prod_pattern =
- "",0,(None,[],Some NotationPt.Binder
+ "",0,(None,[],Some (NotationPt.Binder
(`Pi, (mk_id "_",Some (NotationPt.Appl
[ NotationPt.Implicit `JustOne
; NotationPt.Implicit `JustOne
; NotationPt.UserInput
; NotationPt.Implicit `JustOne ])),
- NotationPt.Implicit `JustOne));;
+ NotationPt.Implicit `JustOne)));;
let prod_pattern_jm =
- "",0,(None,[],Some NotationPt.Binder
+ "",0,(None,[],Some (NotationPt.Binder
(`Pi, (mk_id "_",Some (NotationPt.Appl
[ NotationPt.Implicit `JustOne
; NotationPt.Implicit `JustOne
; NotationPt.UserInput
; NotationPt.Implicit `JustOne
; NotationPt.Implicit `JustOne ])),
- NotationPt.Implicit `JustOne));;
+ NotationPt.Implicit `JustOne)));;
let hp_pattern n =
"",0,(None,[n, NotationPt.Appl
; NotationPt.Implicit `JustOne ] ],
None);;
-(* returns the discrimination = injection+contradiction principle *)
+(* creates the discrimination = injection+contradiction principle *)
+exception ConstructorTooBig of string;;
-let mk_discriminator it ~use_jmeq nleft xyty status =
- let _,indname,_,cl = it in
+let mk_discriminator ~use_jmeq ?(force=false) name it leftno status baseuri =
+ let itnargs =
+ let _,_,arity,_ = it in
+ List.length (arg_list 0 arity) in
+ let _,itname,_,_ = it in
+ let params = List.map (fun x -> "a" ^ string_of_int x) (HExtlib.list_seq 1 (itnargs+1)) in
+ let xyty = mk_appl (List.map mk_id (itname::params)) in
+ (* PHASE 1: derive the type of the discriminator (we'll name it "principle") *)
let mk_eq tys ts us es n =
if use_jmeq then
List.nth us n]
in
+
+ let _,_,_,cl = it in
- let kname it j =
- let _,_,_,cl = it in
+ let kname (*it*) j =
let _,name,_ = List.nth cl j in
name
in
let branch i j ts us =
- let nargs = nargs it nleft i in
+ let nargs = nargs it leftno i in
let es = List.map (fun x -> mk_id ("e" ^ string_of_int x)) (HExtlib.list_seq 0 nargs) in
let tys = List.map
(fun x -> iter
NotationPt.Binder (`Lambda, (mk_id ("p" ^ string_of_int i), None),
acc))) (nargs-1)
(mk_appl [mk_id "eq"; NotationPt.Implicit `JustOne;
- mk_appl (mk_id (kname it i)::
+ mk_appl (mk_id (kname i)::
List.map (fun x -> mk_id ("x" ^string_of_int x))
(HExtlib.list_seq 0 (List.length ts)));
- mk_appl (mk_id (kname it j)::us)])]
+ mk_appl (mk_id (kname j)::us)])]
in
(** NotationPt.Binder (`Lambda, (mk_id "e",
Some (mk_appl
None,
List.map
(fun j ->
- let nargs_kty = nargs it nleft j in
+ let nargs_kty = nargs it leftno j in
let us = iter (fun m acc -> mk_id ("u" ^ (string_of_int m))::acc)
(nargs_kty - 1) [] in
let nones =
iter (fun _ acc -> None::acc) (nargs_kty - 1) [] in
- NotationPt.Pattern (kname it j,
+ NotationPt.Pattern (kname j,
None,
List.combine us nones),
branch i j ts us)
None ,
List.map
(fun i ->
- let nargs_kty = nargs it nleft i in
+ let nargs_kty = nargs it leftno i in
+ if (nargs_kty > 5 && not use_jmeq && not force) then raise (ConstructorTooBig (kname i));
let ts = iter (fun m acc -> mk_id ("t" ^ (string_of_int m))::acc)
(nargs_kty - 1) [] in
let nones =
iter (fun _ acc -> None::acc) (nargs_kty - 1) [] in
- NotationPt.Pattern (kname it i,
+ NotationPt.Pattern (kname i,
None,
List.combine ts nones),
inner i ts)
(HExtlib.list_seq 0 (List.length cl))) in
- let principle = NotationPt.Binder (`Lambda, (mk_id "x", Some xyty),
- NotationPt.Binder (`Lambda, (mk_id "y", Some xyty), outer))
+ let principle =
+ mk_prods params (NotationPt.Binder (`Forall, (mk_id "x",
+ Some xyty),
+ NotationPt.Binder (`Forall, (mk_id "y", Some xyty),
+ (if use_jmeq then
+ NotationPt.Binder (`Forall, (mk_id "e",
+ Some (mk_appl
+ [mk_sym "jmsimeq"; NotationPt.Implicit `JustOne; mk_id "x";
+ NotationPt.Implicit `JustOne; mk_id "y"])),
+ outer)
+ else
+ NotationPt.Binder (`Forall, (mk_id "e",
+ Some (mk_appl [mk_sym "eq";NotationPt.Implicit `JustOne; mk_id "x"; mk_id "y"])),
+ outer)))))
in
pp (lazy ("discriminator = " ^ (NotationPp.pp_term status principle)));
-
- status, principle
-;;
-
-let hd_of_term = function
- | NCic.Appl (hd::_) -> hd
- | t -> t
-;;
-
-let name_of_rel ~context rel =
- let s, _ = List.nth context (rel-1) in s
-;;
-(* let lookup_in_ctx ~context n =
- List.nth context ((List.length context) - n - 1)
-;;*)
-
-let mk_sym s = NotationPt.Symbol (s,0);;
-
-let discriminate_tac ~context cur_eq status =
- pp (lazy (Printf.sprintf "discriminate: equation %s" (name_of_rel ~context cur_eq)));
-
- let dbranch it ~use_jmeq leftno consno =
+ (* PHASE 2: create the object for the proof of the principle: we'll name it
+ * "theorem" *)
+ let status, theorem =
+ let attrs = `Generated, `Theorem, `DiscriminationPrinciple in
+ GrafiteDisambiguate.disambiguate_nobj status ~baseuri
+ (baseuri ^ name ^ ".def",0,
+ NotationPt.Theorem
+ (name, principle, Some (NotationPt.Implicit (`Tagged "inv")), attrs))
+ in
+ let _uri,_height,nmenv,_nsubst,_nobj = theorem in
+ let ninitial_stack = Continuationals.Stack.of_nmetasenv nmenv in
+ let status = status#set_obj theorem in
+ let status = status#set_stack ninitial_stack in
+ let status = subst_metasenv_and_fix_names status in
+
+ (* PHASE 3: we finally prove the discrimination principle *)
+ let dbranch it ~use_jmeq:__ leftno consno =
let refl_id = mk_sym "refl" in
pp (lazy (Printf.sprintf "dbranch %d %d" leftno consno));
let nlist = HExtlib.list_seq 0 (nargs it leftno consno) in
(* (\forall ...\forall P.\forall DH : ( ... = ... -> P). P) *)
let params = List.map (fun x -> NTactics.intro_tac ("a" ^ string_of_int x)) nlist in
- NTactics.reduce_tac ~reduction:(`Normalize true) ~where:default_pattern::
+ (* NTactics.reduce_tac ~reduction:(`Normalize true)
+ * ~where:default_pattern::*)
params @ [
NTactics.intro_tac "P";
NTactics.intro_tac "DH";
] in
let dbranches it ~use_jmeq leftno =
pp (lazy (Printf.sprintf "dbranches %d" leftno));
- let _,_,_,cl = it in
let nbranches = List.length cl in
let branches = iter (fun n acc ->
let m = nbranches - n - 1 in
NTactics.branch_tac ~force:false:: branches @ [NTactics.merge_tac]
else branches
in
+ let print_tac s status = pp s ; status in
+
+ let status =
+ NTactics.block_tac (
+ [print_tac (lazy "ci sono") (*;
+ NTactics.reduce_tac ~reduction:(`Normalize true) ~where:default_pattern *)
+ ]
+ @ List.map (fun x -> NTactics.intro_tac x) params @
+ [NTactics.intro_tac "x";
+ NTactics.intro_tac "y";
+ NTactics.intro_tac "Deq";
+ print_tac (lazy "ci sono 2");
+ NTactics.rewrite_tac ~dir:`RightToLeft ~what:("",0,mk_id "Deq") ~where:default_pattern;
+ NTactics.cases_tac ~what:("",0,mk_id "x") ~where:default_pattern]
+ @ dbranches it ~use_jmeq leftno) status
+ in status, status#obj
+;;
+
+let hd_of_term = function
+ | NCic.Appl (hd::_) -> hd
+ | t -> t
+;;
+
+let name_of_rel ~context rel =
+ let s, _ = List.nth context (rel-1) in s
+;;
+
+(* let lookup_in_ctx ~context n =
+ List.nth context ((List.length context) - n - 1)
+;;*)
+
+let discriminate_tac ~context cur_eq status =
+ pp (lazy (Printf.sprintf "discriminate: equation %s" (name_of_rel ~context cur_eq)));
+
let eq_name,(NCic.Decl s | NCic.Def (s,_)) = List.nth context (cur_eq-1) in
let _,ctx' = HExtlib.split_nth cur_eq context in
let status, s = NTacStatus.whd status ctx' (mk_cic_term ctx' s) in
let status, s = term_of_cic_term status s ctx' in
- let status, leftno, it, use_jmeq =
- let it, t1, t2, use_jmeq = match s with
- | NCic.Appl [_;it;t1;t2] -> it,t1,t2,false
- | NCic.Appl [_;it;t1;_;t2] -> it,t1,t2,true
+ let status,it,use_jmeq =
+ let it,use_jmeq = match s with
+ | NCic.Appl [_;it;_;_] -> it,false
+ | NCic.Appl [_;it;_;_;_] -> it,true
| _ -> assert false in
(* XXX: serve? ho già fatto whd *)
let status, it = whd status ctx' (mk_cic_term ctx' it) in
uri, indtyno, NCicEnvironment.get_checked_indtys status r
| _ -> pp (lazy ("discriminate: indty =" ^ status#ppterm
~metasenv:[] ~subst:[] ~context:[] it)) ; assert false in
- let _,leftno,its,_,_ = its in
- status, leftno, List.nth its indtyno, use_jmeq
+ let _,_,its,_,_ = its in
+ status,List.nth its indtyno, use_jmeq
in
let itnargs =
List.length (arg_list 0 arity) in
let _,itname,_,_ = it in
let params = List.map (fun x -> "a" ^ string_of_int x) (HExtlib.list_seq 1 (itnargs+1)) in
- let xyty = mk_appl (List.map mk_id (itname::params)) in
- let print_tac s status = pp s ; status in
- NTactics.block_tac (
- [(fun status ->
- let status, discr = mk_discriminator it ~use_jmeq leftno xyty status in
- let cut_term = mk_prods params (NotationPt.Binder (`Forall, (mk_id "x",
- Some xyty),
- NotationPt.Binder (`Forall, (mk_id "y", Some xyty),
- (if use_jmeq then fun tgt ->
- NotationPt.Binder (`Forall, (mk_id "e",
- Some (mk_appl
- [mk_sym "jmsimeq"; NotationPt.Implicit `JustOne; mk_id "x";
- NotationPt.Implicit `JustOne; mk_id "y"])),tgt)
- else fun tgt ->
- NotationPt.Binder (`Forall, (mk_id "e",
- Some (mk_appl [mk_sym "eq";NotationPt.Implicit `JustOne; mk_id "x"; mk_id "y"])),tgt))
- (mk_appl [discr; mk_id "x"; mk_id "y"(*;mk_id "e"*)])))) in
- let status = print_tac (lazy ("cut_term = "^ NotationPp.pp_term status cut_term)) status in
- NTactics.cut_tac ("",0, cut_term)
- status);
- NTactics.branch_tac;
- print_tac (lazy "ci sono");
- NTactics.reduce_tac ~reduction:(`Normalize true) ~where:default_pattern
- ]
- @ List.map (fun x -> NTactics.intro_tac x) params @
- [NTactics.intro_tac "x";
- NTactics.intro_tac "y";
- NTactics.intro_tac "Deq";
- print_tac (lazy "ci sono 2");
- NTactics.rewrite_tac ~dir:`RightToLeft ~what:("",0,mk_id "Deq") ~where:default_pattern;
- NTactics.cases_tac ~what:("",0,mk_id "x") ~where:default_pattern]
- @ dbranches it ~use_jmeq leftno @
- [NTactics.shift_tac;
- print_tac (lazy "ci sono 3");
- NTactics.intro_tac "#discriminate";
- NTactics.apply_tac ("",0,mk_appl ([mk_id "#discriminate"]@
+ let principle_name =
+ if use_jmeq then itname ^ "_jmdiscr"
+ else itname ^ "_discr"
+ in
+ pp (lazy ("apply (" ^ principle_name ^ " " ^
+ (String.concat "" (HExtlib.mk_list "?" (List.length params + 2))) ^
+ " " ^ eq_name ^ ")"));
+ NTactics.apply_tac ("",0,mk_appl ([mk_id principle_name]@
HExtlib.mk_list (NotationPt.Implicit `JustOne) (List.length params + 2) @
- [mk_id eq_name ]));
-(* NTactics.reduce_tac ~reduction:(`Normalize true) ~where:default_pattern; *)
- NTactics.clear_tac ["#discriminate"];
- NTactics.merge_tac; print_tac (lazy "the end of discriminate")]
- ) status
+ [mk_id eq_name ])) status
;;
let saturate_skip status context skip =
;;
let subst_tac ~context ~dir skip cur_eq =
- fun status as oldstatus ->
+ fun (status as oldstatus) ->
let eq_name,(NCic.Decl s | NCic.Def (s,_)) = List.nth context (cur_eq-1) in
let _,ctx' = HExtlib.split_nth cur_eq context in
let status, s = NTacStatus.whd status ctx' (mk_cic_term ctx' s) in
| NCic.Rel var ->
cascade_select_in_ctx status ~subst:(get_subst status) context skip (var+cur_eq)
| _ -> cascade_select_in_ctx status ~subst:(get_subst status) context skip cur_eq in
+ let varname = match var with
+ | NCic.Rel var ->
+ let name,_ = List.nth context (var+cur_eq-1) in
+ HLog.warn (Printf.sprintf "destruct: trying to remove variable: %s" name);
+ [name]
+ | _ -> []
+ in
let names_to_gen = List.filter (fun n -> n <> eq_name) names_to_gen in
if (List.exists (fun x -> List.mem x skip) names_to_gen)
then oldstatus
else
- let gen_tac x =
- NTactics.generalize_tac
- ~where:("",0,(Some (mk_id x),[], Some NotationPt.UserInput)) in
- NTactics.block_tac ((List.map gen_tac names_to_gen)@
- [NTactics.clear_tac names_to_gen;
+ let gen_tac x =
+ (fun s ->
+ (*let x' = String.concat " " x in*)
+ let x = List.map mk_id x in
+ (* let s = NTactics.print_tac false ("@generalize " ^ x') s in *)
+ generalize0_tac x s) in
+ NTactics.block_tac (
+ (* (List.map gen_tac names_to_gen)@ *)
+ [gen_tac (List.rev names_to_gen);
+ NTactics.clear_tac names_to_gen;
NTactics.rewrite_tac ~dir
~what:("",0,mk_id eq_name) ~where:default_pattern;
(* NTactics.reduce_tac ~reduction:(`Normalize true)
~where:default_pattern;*)
- NTactics.try_tac (NTactics.clear_tac [eq_name])]@
+ (* XXX: temo che la clear multipla funzioni bene soltanto se
+ * gli identificatori sono nell'ordine giusto.
+ * Per non saper né leggere né scrivere, usiamo due clear
+ * invece di una *)
+ NTactics.try_tac (NTactics.clear_tac [eq_name]);
+ NTactics.try_tac (NTactics.clear_tac varname);
+]@
(List.map NTactics.intro_tac (List.rev names_to_gen))) status
;;
let status, s = NTacStatus.whd status ctx' (mk_cic_term ctx' s) in
let status, s = term_of_cic_term status s ctx' in
let skip = saturate_skip status context skip in
- (*
- let streicher_id =
- match s with
- | NCic.Appl [_;_;_;_] -> mk_id "streicherK"
- | NCic.Appl [_;_;_;_;_] -> mk_id "streicherKjmeq"
- | _ -> assert false
- in
- pp (lazy (Printf.sprintf "clearid: equation %s" eq_name));
- let names_to_gen, _ =
- cascade_select_in_ctx ~subst:(get_subst status) context cur_eq in
- let names_to_gen = names_to_gen @ [eq_name] in
- let gen_tac x =
- NTactics.generalize_tac
- ~where:("",0,(Some (mk_id x),[], Some NotationPt.UserInput)) in
- NTactics.block_tac ((List.map gen_tac names_to_gen)@
- [NTactics.clear_tac names_to_gen;
- NTactics.apply_tac ("",0, mk_appl [streicher_id;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne]);
- NTactics.reduce_tac ~reduction:(`Normalize true)
- ~where:default_pattern] @
- (let names_to_intro =
- match List.rev names_to_gen with
- | [] -> []
- | _::tl -> tl in
- List.map NTactics.intro_tac names_to_intro)) status
-*)
pp (lazy (Printf.sprintf "clearid: equation %s" eq_name));
- match s with
- | NCic.Appl [_;_;_;_] ->
- (* leibniz *)
- let streicher_id = mk_id "streicherK"
- in
- let names_to_gen, _ =
- cascade_select_in_ctx status ~subst:(get_subst status) context skip cur_eq in
- let names_to_gen = names_to_gen @ [eq_name] in
- let gen_tac x =
- NTactics.generalize_tac
- ~where:("",0,(Some (mk_id x),[], Some NotationPt.UserInput)) in
- NTactics.block_tac ((List.map gen_tac names_to_gen)@
- [NTactics.clear_tac names_to_gen;
- NTactics.apply_tac ("",0, mk_appl [streicher_id;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne]);
-(* NTactics.reduce_tac ~reduction:(`Normalize true)
- ~where:default_pattern *)
- ] @
- (let names_to_intro =
- match List.rev names_to_gen with
- | [] -> []
- | _::tl -> tl in
- List.map NTactics.intro_tac names_to_intro)) status
- | NCic.Appl [_;_;_;_;_] ->
- (* JMeq *)
- let streicher_id = mk_id "streicherK"
- in
- let names_to_gen, _ =
- cascade_select_in_ctx status ~subst:(get_subst status) context skip cur_eq in
- let names_to_gen = names_to_gen (* @ [eq_name]*) in
- let gen_tac x =
- NTactics.generalize_tac
- ~where:("",0,(Some (mk_id x),[], Some NotationPt.UserInput)) in
- let gen_eq = NTactics.generalize_tac
- ~where:("",0,(Some (mk_appl [mk_id "jmeq_to_eq";
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne;
- mk_id eq_name]),[], Some NotationPt.UserInput)) in
- NTactics.block_tac ((List.map gen_tac names_to_gen)@gen_eq::
- [NTactics.clear_tac names_to_gen;
- NTactics.try_tac (NTactics.clear_tac [eq_name]);
- NTactics.apply_tac ("",0, mk_appl [streicher_id;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne;
- NotationPt.Implicit `JustOne]);
-(* NTactics.reduce_tac ~reduction:(`Normalize true)
- ~where:default_pattern *)
- ] @
- (let names_to_intro = List.rev names_to_gen in
- List.map NTactics.intro_tac names_to_intro)) status
- | _ -> assert false
+ let streicher_id = mk_id "streicherK" in
+ let names_to_gen, _ =
+ cascade_select_in_ctx status ~subst:(get_subst status) context skip cur_eq in
+ let gen_tac x = generalize0_tac (List.map mk_id x) in
+
+ match s with
+ (* jmeq *)
+ | NCic.Appl [_;_;_;_;_] ->
+ let names_to_gen = List.rev names_to_gen in
+ (*let gen_eq = NTactics.generalize_tac
+ ~where:("",0,(Some (mk_appl [mk_id "jmeq_to_eq";
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne;
+ mk_id eq_name]),[], Some
+ NotationPt.UserInput)) in*)
+ let gen_eq = generalize0_tac
+ [mk_appl [mk_id "jmeq_to_eq";
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne;
+ mk_id eq_name]] in
+ NTactics.block_tac ((gen_tac (List.rev names_to_gen))::gen_eq::
+ [NTactics.clear_tac names_to_gen;
+ NTactics.try_tac (NTactics.clear_tac [eq_name]);
+ NTactics.apply_tac ("",0, mk_appl [streicher_id;
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne]);
+ ] @
+ (List.map NTactics.intro_tac names_to_gen)) status
+
+ (* leibniz *)
+ | NCic.Appl [_;_;_;_] ->
+ begin
+ let names_to_gen, _ =
+ cascade_select_in_ctx status ~subst:(get_subst status) context skip cur_eq in
+ let names_to_gen = eq_name :: (List.rev names_to_gen) in
+ NTactics.block_tac ((gen_tac names_to_gen)::
+ [NTactics.clear_tac names_to_gen;
+ NTactics.apply_tac ("",0, mk_appl [streicher_id;
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne;
+ NotationPt.Implicit `JustOne])
+ (* NTactics.reduce_tac ~reduction:(`Normalize true)
+ ~where:default_pattern *)
+ ] @
+ let names_to_intro = List.tl names_to_gen in
+ (List.map NTactics.intro_tac names_to_intro)) status
+ end
+ | _ -> assert false
+
;;
let get_ctx st goal =
| _ -> status, `NonEq
in match kind with
| `Identity ->
- let status, goalty = term_of_cic_term status (get_goalty status goal) ctx in
+ let status, _goalty = term_of_cic_term status (get_goalty status goal) ctx in
status, Some (List.length ctx - i), kind
| `Cycle | `Blob | `NonEq -> aux (i+1) (* XXX: skip cyclic/blob equations for now *)
| _ ->
let has_cleared =
try
let _ = NTactics.find_in_context eq_name (get_ctx status' newgoal) in false
- with _ -> true in
+ with
+ | Sys.Break as e -> raise e
+ |_ -> true in
let rm_eq b l = if b then List.filter (fun x -> x <> eq_name) l else l in
let acc = rm_eq has_cleared acc in
let skip = rm_eq has_cleared skip in
let has_cleared =
try
let _ = NTactics.find_in_context eq_name (get_ctx status' newgoal) in false
- with _ -> true in
+ with
+ | Sys.Break as e -> raise e
+ | _ -> true in
let rm_eq b l = if b then List.filter (fun x -> x <> eq_name) l else l in
let acc = rm_eq has_cleared acc in
let skip = rm_eq has_cleared skip in
projects / helm.git / blobdiff