6 let debug_display_arities = false;;
8 (************ Syntax ************************************)
12 (* Var n = n-th De Bruijn index, 0-based *)
21 type var = int * (* arity of variable*) int;;
22 type 'nf_nob i_var_ = [ `I of var * 'nf_nob Listx.listx | `Var of var ]
23 type 'nf_nob i_n_var_ = [ `N of int | 'nf_nob i_var_ ]
24 type ('nf_nob,'nf) i_num_var_ = [
26 | `Match of ('nf_nob,'nf) i_num_var_ * (* originating var *) var * (*lift*) int * (*branches*)(int * 'nf) list ref * (*args*)'nf_nob list
28 type 'nf nf_nob_ = [ `Lam of (* was_unpacked *) bool * 'nf | `Pacman | ('nf nf_nob_,'nf) i_num_var_ ]
29 type nf = [ nf nf_nob_ | `Bottom ]
30 type nf_nob = nf nf_nob_
31 type i_var = nf_nob i_var_;;
32 type i_n_var = nf_nob i_n_var_;;
33 type i_num_var = (nf_nob,nf) i_num_var_;;
45 | `Match _ -> assert false
54 let aux_var l (n, ar) = (if n < l then n else n+m), ar in
55 let rec aux_i_num_var l =
57 `I(v,args) -> `I(aux_var l v, Listx.map (aux_nob l) args)
58 | `Var v -> `Var(aux_var l v)
60 | `Match(t,v,lift,bs,args) ->
61 `Match(aux_i_num_var l t, v, lift + m, bs, List.map (aux_nob l) args)
64 #i_num_var as x -> (aux_i_num_var l x :> nf_nob)
65 | `Lam(b,nf) -> `Lam (b, aux (l+1) nf)
69 #nf_nob as x -> (aux_nob l x :> nf)
75 (* put t under n lambdas, lifting t accordingtly *)
79 | n when n > 0 -> `Lam (false, lift 1 (make_lams t (n-1)))
83 let rec aux n = function
85 | `Var(x,ar) -> if x < n then [] else [(x-n,ar)]
87 (if x < n then [] else [(x-n,ar)]) @
88 List.concat (List.map (aux n) (Listx.to_list args :> nf list))
89 | `Lam(_,t) -> aux (n+1) t
90 | `Match(t,_,liftno,bs,args) ->
92 List.concat (List.map (fun (_,t) -> aux (n-liftno) t) !bs) @
93 List.concat (List.map (aux n) (args :> nf list))
94 | `Bottom | `Pacman -> []
97 let free_vars = (List.map fst) ++ free_vars';;
102 let rec scott_of_nf = function
103 | `N n -> Scott.mk_n n
104 | `Var(v,_) -> Pure.V v
105 | `Match(t,_,liftno,bs,args) ->
106 let bs = List.map (fun (n,t) -> n, scott_of_nf (lift liftno (t :> nf))) !bs in
107 let t = scott_of_nf (t :> nf) in
108 let m = Scott.mk_match t bs in
109 List.fold_left (fun acc t -> Pure.A(acc,scott_of_nf t)) m (args :> nf list)
110 | `I((v,_), args) -> Listx.fold_left (fun acc t -> Pure.A(acc,scott_of_nf t)) (Pure.V v) (args :> nf Listx.listx)
111 | `Lam(_,t) -> Pure.L (scott_of_nf t)
113 | `Pacman -> let f x = Pure.A (x,x) in f (Pure.L (Pure.L (f (Pure.V 0))))
117 (************ Pretty-printing ************************************)
119 (* let rec string_of_term l = fun _ -> "";; *)
121 let rec string_of_term l =
122 let rec string_of_term_w_pars l = function
123 | `Var(n,ar) -> print_name l n ^ (if debug_display_arities then ":" ^ string_of_int ar else "")
124 | `N n -> string_of_int n
125 | `I _ as t -> "(" ^ string_of_term_no_pars_app l t ^ ")"
126 | `Lam(_,`Bottom) -> "BOMB"
127 | `Lam _ as t -> "(" ^ string_of_term_no_pars_lam l t ^ ")"
128 | `Match(t,(v,ar),bs_lift,bs,args) ->
129 "[match_"^ string_of_var v ^ (if debug_display_arities then ":"^ string_of_int ar else "") ^"_ " ^ string_of_term_no_pars l (t :> nf) ^
130 " with " ^ String.concat " | " (List.map (fun (n,t) -> string_of_int n ^ " => " ^ string_of_term l (lift bs_lift (t :> nf))) !bs) ^ "] " ^
131 String.concat " " (List.map (string_of_term l) (args :> nf list)) ^ ")"
134 and string_of_term_no_pars_app l = function
135 | `I((n,ar), args) -> print_name l n ^ (if debug_display_arities then ":" ^ string_of_int ar else "") ^ " " ^ String.concat " " (List.map (string_of_term_w_pars l) (Listx.to_list args :> nf list))
136 | #nf as t -> string_of_term_w_pars l t
137 and string_of_term_no_pars_lam l = function
138 | `Lam(_,`Bottom) -> "BOMB"
139 | `Lam(_,t) -> let name = string_of_var (List.length l) in
140 "λ" ^ name ^ ". " ^ (string_of_term_no_pars_lam (name::l) t)
141 | _ as t -> string_of_term_no_pars l t
142 and string_of_term_no_pars l = function
143 | `Lam _ as t -> string_of_term_no_pars_lam l t
144 | #nf as t -> string_of_term_no_pars_app l t
145 in string_of_term_no_pars l
148 let print ?(l=[]) = string_of_term l;;
149 let string_of_nf t = string_of_term [] (t :> nf);;
151 (************ Hereditary substitutions ************************************)
155 #i_var as y -> (y : i_var)
157 prerr_endline (print (t :> nf));
158 assert false (* algorithm failed *)
160 let cast_to_i_n_var =
162 #i_n_var as y -> (y : i_n_var)
164 prerr_endline (print (t :> nf));
165 assert false (* algorithm failed *)
167 let cast_to_i_num_var =
169 #i_num_var as y -> (y : i_num_var)
171 prerr_endline (print (t :> nf));
172 assert false (* algorithm failed *)
174 let rec set_arity arity = function
175 | `Var(n,_) -> `Var(n,arity)
176 | `N _ | `Bottom | `Pacman as t -> t
177 | `Lam(false, t) -> `Lam(false, set_arity arity t)
178 | `Match(t,(n,_),bs_lift,bs,args) -> `Match(t,(n,arity),bs_lift,bs,args)
179 | `I _ | `Lam _ -> assert false
181 let minus1 n = if n = min_int then n else n - 1;;
183 let rec mk_app (h : nf) (arg : nf) =
188 | `I(v,args) -> `I(v,Listx.append (Listx.Nil arg) args)
189 | `Var v -> `I(v, Listx.Nil arg)
190 | `Lam(truelam,nf) -> subst truelam true 0 arg (nf : nf) (* AC FIXME sanity check on arity *)
191 | `Match(t,v,lift,bs,args) -> `Match(t,v,lift,bs,List.append args [arg])
192 | `Bottom | `Pacman as t -> t
193 | `N _ -> assert false (* Numbers cannot be applied *)
194 (*in let l = ["v0";"v1";"v2"] in
195 prerr_endline ("mk_app h:" ^ print ~l h ^ " arg:" ^ print ~l:l arg ^ " res:" ^ print ~l:l res); res*)
198 (*prerr_endline ("MK_APPL: " ^ print h ^ " " ^ String.concat " " (List.map print args));*)
199 List.fold_left mk_app h args
201 and mk_appx h args = Listx.fold_left mk_app h args
203 and mk_match t (n,ar) bs_lift bs args =
204 (*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));*)
209 let h = List.assoc m !bs in
210 let h = set_arity (minus1 ar) h in
211 let h = lift bs_lift h in
214 `Match (t,(n,ar),bs_lift,bs,[]))
215 (* We are assuming that the econding of matches is s.t.:
216 - match PAC.. --> PAC
217 - match BOT.. --> BOT *)
220 | `Lam _ -> assert false
221 | `I _ | `Var _ | `Match _ as t -> `Match(t,(n,ar),bs_lift,bs,[]) in
224 and subst truelam delift_by_one what (with_what : nf(*_nob*)) (where : nf) =
225 let rec aux_propagate_arity ar = function
226 | `Lam(false, t) when not delift_by_one -> `Lam(false, aux_propagate_arity ar t)
227 | `Match(`I(v,args),(x,_),liftno,bs,args') when not delift_by_one ->
228 `Match(`I(v,args),(x,ar),liftno,bs,args')
229 | `Var(i,oldar) -> `Var(i, if truelam then (assert (oldar = min_int); ar) else oldar)
231 let rec aux_i_num_var l =
235 let args = Listx.map (aux l) (args :> nf Listx.listx) in
236 mk_appx (lift l (aux_propagate_arity ar (with_what :> nf))) args
238 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)))
241 lift l (aux_propagate_arity ar (with_what :> nf))
243 `Var((if delift_by_one && n >= l then n-1 else n), ar)
245 | `Match(t,v,bs_lift,bs,args) ->
246 let bs_lift = bs_lift + if delift_by_one then -1 else 0 in
247 let l' = l - bs_lift in
248 let with_what' = lift l' (with_what :> nf) in
249 (* The following line should be the identity when delift_by_one = true because we
250 are assuming the ts to not contain lambda-bound variables. *)
251 bs := List.map (fun (n,t) -> n,subst truelam false what with_what' t) !bs ;
252 let body = aux_i_num_var l t in
253 mk_match body v bs_lift bs (List.map (aux l) (args :> nf list))
255 (*function iii -> let res = match iii with*)
257 | #i_num_var as x -> aux_i_num_var l x
258 | `Lam(b, nf) -> `Lam(b, aux (l+1) nf)
261 (*in let ll = ["v0";"v1";"v2"] in
262 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*)
267 (************ Parsing ************************************)
270 let fix_arity = function
271 | `I((n,_),args) -> `I((n,1+Listx.length args),args)
272 | _ -> assert false in
273 let (tms, free) = Parser.parse_many strs in
274 (* Replace pacmans and bottoms *)
275 let n_bot = try Util.index_of "BOT" free with Not_found -> min_int in
276 let n_pac = try Util.index_of "PAC" free with Not_found -> min_int in
277 let n_bomb = try Util.index_of "BOMB" free with Not_found -> min_int in
279 if v = lev + n_bot then `Bottom
280 else if v = lev + n_pac then `Pacman
281 else if v = lev + n_bomb then `Lam(true, `Bottom)
282 else `Var(v,1) in (* 1 by default when variable not applied *)
284 let rec aux lev = function
285 | Parser.Lam t -> `Lam (true, aux (lev+1) t)
286 | Parser.App (t1, t2) -> fix_arity (mk_app (aux lev t1) (aux lev t2))
287 | Parser.Var v -> fix lev v in
288 List.map (aux 0) tms, free
291 (************** Algorithm(s) ************************)
293 let eta_compare x y =
294 (* let clex a b = let diff = ? a b in if diff = 0 then cont () else 0 in *)
295 let clex aux1 aux2 (a1,a2) (b1,b2) =
296 let diff = aux1 a1 b1 in if diff = 0 then aux2 a2 b2 else diff in
297 let rec lex aux l1 l2 =
302 | x::xs, y::ys -> clex aux (lex aux) (x,xs) (y,ys) in
303 let rec aux t1 t2 = match t1, t2 with
304 | `Var(n,_) , `Var(m,_) -> compare n m
305 | `I((n1,_), l1), `I((n2,_), l2) ->
306 clex compare (lex aux) (n1, (Listx.to_list l1 :> nf list)) (n2, (Listx.to_list l2 :> nf list))
308 | `Pacman, `Pacman -> 0
312 | `Lam _, `Bottom -> assert false (* TO BE UNDERSTOOD *)
313 | `Lam(_,t1), `Lam(_,t2) -> aux t1 t2
314 | `Lam(_,t1), t2 -> - aux t1 (mk_app (lift 1 t2) (`Var(0,-666)))
315 | t2, `Lam(_,t1) -> aux t1 (mk_app (lift 1 t2) (`Var(0,-666)))
316 | `N n1, `N n2 -> compare n1 n2
317 | `Match(u,_,bs_lift,bs,args), `Match(u',_,bs_lift',bs',args') ->
318 let bs = List.sort (fun (n,_) (m,_) -> compare n m) !bs in
319 let bs' = List.sort (fun (n,_) (m,_) -> compare n m) !bs' in
320 clex aux (clex (lex (clex compare aux)) (lex aux)) ((u :> nf), (bs, (args :> nf list))) ((u' :> nf), (bs', (args' :> nf list)))
334 let eta_eq (#nf as x) (#nf as y) = 0 = eta_compare x y ;;
336 let rec eta_subterm sub t =
337 if eta_eq sub t then true else
339 | `Lam(_,t') -> eta_subterm (lift 1 sub) t'
342 | `Match(u,ar,liftno,bs,args) ->
343 eta_subterm sub (u :> nf)
344 || List.exists (fun (_, t) -> eta_subterm sub (lift liftno t)) !bs
345 || List.exists (eta_subterm sub) (args :> nf list)
346 | `I(v, args) -> List.exists (eta_subterm sub) ((Listx.to_list args) :> nf list) || (match sub with
348 | `I(v', args') -> v = v'
349 && Listx.length args' < Listx.length args
350 && 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))
353 | `N _ | `Var _ -> false
356 let eta_subterm (#nf as x) (#nf as y) = eta_subterm x y;;
358 let max_arity_tms n =
359 let max a b = match a, b with
362 | Some x, None -> Some x
363 | Some x, Some y -> Some (Pervasives.max x y) in
364 let aux_var l (m,a) = if n + l = m then Some a else None in
365 let rec aux l = function
366 | `Var v -> aux_var l v
367 | `I(v,tms) -> max (aux_var l v) (aux_tms l (Listx.to_list tms :> nf list))
368 | `Lam(_,t) -> aux (l+1) t
369 | `Match(u,_,_,bs,args) -> max (max (aux l (u :> nf)) (aux_tms l (args :> nf list))) (aux_tms l (List.map snd !bs))
370 | `N _ | `Bottom | `Pacman -> None
372 List.fold_left (fun acc t -> Pervasives.max acc (aux l t)) None in
373 fun tms -> aux_tms 0 (tms :> nf list)