5 (************ Syntax ************************************)
9 (* Var n = n-th De Bruijn index, 0-based *)
18 type var = int * (* arity of variable*) int;;
19 type 'nf_nob i_var_ = [ `I of var * 'nf_nob Listx.listx | `Var of var ]
20 type 'nf_nob i_n_var_ = [ `N of int | 'nf_nob i_var_ ]
21 type ('nf_nob,'nf) i_num_var_ = [
23 | `Match of ('nf_nob,'nf) i_num_var_ * (* originating var *) var * (*lift*) int * (*branches*)(int * 'nf) list ref * (*args*)'nf_nob list
25 type 'nf nf_nob_ = [ `Lam of (* was_unpacked *) bool * 'nf | `Pacman | ('nf nf_nob_,'nf) i_num_var_ ]
26 type nf = [ nf nf_nob_ | `Bottom ]
27 type nf_nob = nf nf_nob_
28 type i_var = nf_nob i_var_;;
29 type i_n_var = nf_nob i_n_var_;;
30 type i_num_var = (nf_nob,nf) i_num_var_;;
42 | `Match _ -> assert false
51 let aux_var l (n, ar) = (if n < l then n else n+m), ar in
52 let rec aux_i_num_var l =
54 `I(v,args) -> `I(aux_var l v, Listx.map (aux_nob l) args)
55 | `Var v -> `Var(aux_var l v)
57 | `Match(t,v,lift,bs,args) ->
58 `Match(aux_i_num_var l t, v, lift + m, bs, List.map (aux_nob l) args)
61 #i_num_var as x -> (aux_i_num_var l x :> nf_nob)
62 | `Lam(b,nf) -> `Lam (b, aux (l+1) nf)
66 #nf_nob as x -> (aux_nob l x :> nf)
72 (* put t under n lambdas, lifting t accordingtly *)
76 | n when n > 0 -> `Lam (false, lift 1 (make_lams t (n-1)))
80 let rec aux n = function
82 | `Var(x,ar) -> if x < n then [] else [(x-n,ar)]
84 (if x < n then [] else [(x-n,ar)]) @
85 List.concat (List.map (aux n) (Listx.to_list args :> nf list))
86 | `Lam(_,t) -> aux (n+1) t
87 | `Match(t,_,liftno,bs,args) ->
89 List.concat (List.map (fun (_,t) -> aux (n-liftno) t) !bs) @
90 List.concat (List.map (aux n) (args :> nf list))
91 | `Bottom | `Pacman -> []
94 let free_vars = (List.map fst) ++ free_vars';;
101 | `N n -> Scott.mk_n n
102 | `Var(v,_) -> Pure.V v
103 | `Match(t,_,liftno,bs,args) ->
104 let bs = List.map (fun (n,t) -> n, scott_of_nf (lift liftno (t :> nf))) !bs in
105 let t = scott_of_nf (t :> nf) in
106 let m = Scott.mk_match t bs in
107 List.fold_left (fun acc t -> Pure.A(acc,scott_of_nf t)) m (args :> nf list)
108 | `I((v,_), args) -> Listx.fold_left (fun acc t -> Pure.A(acc,scott_of_nf t)) (Pure.V v) (args :> nf Listx.listx)
109 | `Lam(b,f) -> Pure.L (scott_of_nf f)
110 | `Bottom -> let f x = Pure.A (x,x) in f (Pure.L (f (Pure.V 0)))
111 | `Pacman -> let f x = Pure.A (x,x) in f (Pure.L (Pure.L (f (Pure.V 0))))
115 (************ Pretty-printing ************************************)
117 (* let rec string_of_term l = fun _ -> "";; *)
119 let rec string_of_term l =
120 let rec string_of_term_w_pars l = function
121 | `Var(n,ar) -> print_name l n ^ ":" ^ string_of_int ar
122 | `N n -> string_of_int n
123 | `I _ as t -> "(" ^ string_of_term_no_pars_app l t ^ ")"
124 | `Lam _ as t -> "(" ^ string_of_term_no_pars_lam l t ^ ")"
125 | `Match(t,(v,ar),bs_lift,bs,args) ->
126 "[match("^ string_of_var v ^":"^ string_of_int ar ^") " ^ string_of_term_no_pars l (t :> nf) ^
127 " with " ^ String.concat " | " (List.map (fun (n,t) -> string_of_int n ^ " => " ^ string_of_term l (lift bs_lift (t :> nf))) !bs) ^ "] " ^
128 String.concat " " (List.map (string_of_term l) (args :> nf list)) ^ ")"
131 and string_of_term_no_pars_app l = function
132 | `I((n,ar), args) -> print_name l n ^ ":" ^ string_of_int ar ^ " " ^ String.concat " " (List.map (string_of_term_w_pars l) (Listx.to_list args :> nf list))
133 | #nf as t -> string_of_term_w_pars l t
134 and string_of_term_no_pars_lam l = function
135 | `Lam(_,t) -> let name = string_of_var (List.length l) in
136 "λ" ^ name ^ ". " ^ (string_of_term_no_pars_lam (name::l) t)
137 | _ as t -> string_of_term_no_pars l t
138 and string_of_term_no_pars l = function
139 | `Lam _ as t -> string_of_term_no_pars_lam l t
140 | #nf as t -> string_of_term_no_pars_app l t
141 in string_of_term_no_pars l
144 let print ?(l=[]) = string_of_term l;;
145 let string_of_nf t = string_of_term [] (t :> nf);;
147 (************ Hereditary substitutions ************************************)
151 #i_var as y -> (y : i_var)
153 prerr_endline (print (t :> nf));
154 assert false (* algorithm failed *)
156 let cast_to_i_n_var =
158 #i_n_var as y -> (y : i_n_var)
160 prerr_endline (print (t :> nf));
161 assert false (* algorithm failed *)
163 let cast_to_i_num_var =
165 #i_num_var as y -> (y : i_num_var)
167 prerr_endline (print (t :> nf));
168 assert false (* algorithm failed *)
170 let rec set_arity arity = function
171 | `Var(n,_) -> `Var(n,arity)
172 | `N _ | `Bottom | `Pacman as t -> t
173 | `Lam(false, t) -> `Lam(false, set_arity arity t)
174 | `Match(t,(n,_),bs_lift,bs,args) -> `Match(t,(n,arity),bs_lift,bs,args)
175 | `I _ | `Lam _ -> assert false
177 let minus1 n = if n = min_int then n else n - 1;;
179 let rec mk_app (h : nf) (arg : nf) =
184 | `I(v,args) -> `I(v,Listx.append (Listx.Nil arg) args)
185 | `Var v -> `I(v, Listx.Nil arg)
186 | `Lam(truelam,nf) -> subst truelam true 0 arg (nf : nf) (* AC FIXME sanity check on arity *)
187 | `Match(t,v,lift,bs,args) -> `Match(t,v,lift,bs,List.append args [arg])
188 | `Bottom | `Pacman as t -> t
189 | `N _ -> assert false (* Numbers cannot be applied *)
190 (*in let l = ["v0";"v1";"v2"] in
191 prerr_endline ("mk_app h:" ^ print ~l h ^ " arg:" ^ print ~l:l arg ^ " res:" ^ print ~l:l res); res*)
194 (*prerr_endline ("MK_APPL: " ^ print h ^ " " ^ String.concat " " (List.map print args));*)
195 List.fold_left mk_app h args
197 and mk_appx h args = Listx.fold_left mk_app h args
199 and mk_match t (n,ar) bs_lift bs args =
200 (*prerr_endline ("MK_MATCH: ([" ^ print t ^ "] " ^ String.concat " " (Listx.to_list (Listx.map (fun (n,t) -> string_of_int n ^ " => " ^ print t) bs)) ^ ") " ^ String.concat " " (List.map print args));*)
205 let h = List.assoc m !bs in
206 let h = set_arity (minus1 ar) h in
207 let h = lift bs_lift h in
210 `Match (t,(n,ar),bs_lift,bs,[]))
211 (* We are assuming that the econding of matches is s.t.:
212 - match PAC.. --> PAC
213 - match BOT.. --> BOT *)
216 | `Lam _ -> assert false
217 | `I _ | `Var _ | `Match _ as t -> `Match(t,(n,ar),bs_lift,bs,[]) in
220 and subst truelam delift_by_one what (with_what : nf(*_nob*)) (where : nf) =
221 let rec aux_propagate_arity ar = function
222 | `Lam(false, t) when not delift_by_one -> `Lam(false, aux_propagate_arity ar t)
223 | `Match(`I(v,args),(x,_),liftno,bs,args') when not delift_by_one ->
224 `Match(`I(v,args),(x,ar),liftno,bs,args')
225 | `Var(i,oldar) -> `Var(i, if truelam then (assert (oldar = min_int); ar) else oldar)
227 let rec aux_i_num_var l =
231 let args = Listx.map (aux l) (args :> nf Listx.listx) in
232 mk_appx (lift l (aux_propagate_arity ar (with_what :> nf))) args
234 mk_appl (`Var ((if delift_by_one && n >= l then n-1 else n), ar)) (List.map (aux l) (Listx.to_list (args :> nf Listx.listx)))
237 lift l (aux_propagate_arity ar (with_what :> nf))
239 `Var((if delift_by_one && n >= l then n-1 else n), ar)
241 | `Match(t,v,bs_lift,bs,args) ->
242 let bs_lift = bs_lift + if delift_by_one then -1 else 0 in
243 let l' = l - bs_lift in
244 let with_what' = lift l' (with_what :> nf) in
245 (* The following line should be the identity when delift_by_one = true because we
246 are assuming the ts to not contain lambda-bound variables. *)
247 bs := List.map (fun (n,t) -> n,subst truelam false what with_what' t) !bs ;
248 let body = aux_i_num_var l t in
249 mk_match body v bs_lift bs (List.map (aux l) (args :> nf list))
251 (*function iii -> let res = match iii with*)
253 | #i_num_var as x -> aux_i_num_var l x
254 | `Lam(b, nf) -> `Lam(b, aux (l+1) nf)
257 (*in let ll = ["v0";"v1";"v2"] in
258 prerr_endline ("subst l:" ^ string_of_int l ^ " delift_by_one:" ^ string_of_bool delift_by_one ^ " what:" ^ (List.nth ll what) ^ " with_what:" ^ print ~l:ll with_what ^ " where:" ^ print ~l:ll iii ^ " res:" ^ print ~l:ll res); res*)
263 (************ Parsing ************************************)
266 let fix_arity = function
267 | `I((n,_),args) -> `I((n,1+Listx.length args),args)
268 | _ -> assert false in
269 let (tms, free) = Parser.parse_many strs in
270 (* Replace pacmans and bottoms *)
271 let n_bot = try Util.index_of "BOT" free with Not_found -> min_int in
272 let n_pac = try Util.index_of "PAC" free with Not_found -> min_int in
274 if v = lev + n_bot then `Bottom
275 else if v = lev + n_pac then `Pacman
276 else `Var(v,1) in (* 1 by default when variable not applied *)
278 let rec aux lev = function
279 | Parser.Lam t -> `Lam (true, aux (lev+1) t)
280 | Parser.App (t1, t2) -> fix_arity (mk_app (aux lev t1) (aux lev t2))
281 | Parser.Var v -> fix lev v in
282 List.map (aux 0) tms, free
285 (************** Algorithm(s) ************************)
287 let eta_compare x y =
288 (* let clex a b = let diff = ? a b in if diff = 0 then cont () else 0 in *)
289 let clex aux1 aux2 (a1,a2) (b1,b2) =
290 let diff = aux1 a1 b1 in if diff = 0 then aux2 a2 b2 else diff in
291 let rec lex aux l1 l2 =
296 | x::xs, y::ys -> clex aux (lex aux) (x,xs) (y,ys) in
297 let rec aux t1 t2 = match t1, t2 with
298 | `Var(n,_) , `Var(m,_) -> compare n m
299 | `I((n1,_), l1), `I((n2,_), l2) ->
300 clex compare (lex aux) (n1, (Listx.to_list l1 :> nf list)) (n2, (Listx.to_list l2 :> nf list))
302 | `Pacman, `Pacman -> 0
306 | `Lam _, `Bottom -> assert false (* TO BE UNDERSTOOD *)
307 | `Lam(_,t1), `Lam(_,t2) -> aux t1 t2
308 | `Lam(_,t1), t2 -> - aux t1 (mk_app (lift 1 t2) (`Var(0,-666)))
309 | t2, `Lam(_,t1) -> aux t1 (mk_app (lift 1 t2) (`Var(0,-666)))
310 | `N n1, `N n2 -> compare n1 n2
311 | `Match(u,_,bs_lift,bs,args), `Match(u',_,bs_lift',bs',args') ->
312 let bs = List.sort (fun (n,_) (m,_) -> compare n m) !bs in
313 let bs' = List.sort (fun (n,_) (m,_) -> compare n m) !bs' in
314 clex aux (clex (lex (clex compare aux)) (lex aux)) ((u :> nf), (bs, (args :> nf list))) ((u' :> nf), (bs', (args' :> nf list)))
328 let eta_eq (#nf as x) (#nf as y) = 0 = eta_compare x y ;;
330 let rec eta_subterm sub t =
331 if eta_eq sub t then true else
333 | `Lam(_,t') -> eta_subterm (lift 1 sub) t'
336 | `Match(u,ar,liftno,bs,args) ->
337 eta_subterm sub (u :> nf)
338 || List.exists (fun (_, t) -> eta_subterm sub (lift liftno t)) !bs
339 || List.exists (eta_subterm sub) (args :> nf list)
340 | `I(v, args) -> List.exists (eta_subterm sub) ((Listx.to_list args) :> nf list) || (match sub with
342 | `I(v', args') -> v = v'
343 && Listx.length args' < Listx.length args
344 && List.for_all (fun (x,y) -> eta_eq x y) (List.combine (Util.take (Listx.length args') (Listx.to_list args :> nf list)) (Listx.to_list args' :> nf list))
347 | `N _ | `Var _ -> false
350 let eta_subterm (#nf as x) (#nf as y) = eta_subterm x y;;
352 let max_arity_tms n =
353 let max a b = match a, b with
356 | Some x, None -> Some x
357 | Some x, Some y -> Some (Pervasives.max x y) in
358 let aux_var l (m,a) = if n + l = m then Some a else None in
359 let rec aux l = function
360 | `Var v -> aux_var l v
361 | `I(v,tms) -> max (aux_var l v) (aux_tms l (Listx.to_list tms :> nf list))
362 | `Lam(_,t) -> aux (l+1) t
363 | `Match(u,_,_,bs,args) -> max (max (aux l (u :> nf)) (aux_tms l (args :> nf list))) (aux_tms l (List.map snd !bs))
364 | `N _ | `Bottom | `Pacman -> None
366 List.fold_left (fun acc t -> Pervasives.max acc (aux l t)) None in
367 fun tms -> aux_tms 0 (tms :> nf list)