0
|
1 |
(* Title: pattern
|
|
2 |
ID: $Id$
|
|
3 |
Author: Tobias Nipkow and Christine Heinzelmann, TU Muenchen
|
|
4 |
Copyright 1993 TU Muenchen
|
|
5 |
|
|
6 |
Unification of Higher-Order Patterns.
|
|
7 |
|
|
8 |
See also:
|
|
9 |
Tobias Nipkow. Functional Unification of Higher-Order Patterns.
|
|
10 |
In Proceedings of the 8th IEEE Symposium Logic in Computer Science, 1993.
|
|
11 |
*)
|
|
12 |
|
|
13 |
signature PATTERN =
|
|
14 |
sig
|
|
15 |
type type_sig
|
|
16 |
type sg
|
|
17 |
type env
|
|
18 |
val eta_contract: term -> term
|
|
19 |
val match: type_sig -> term * term
|
|
20 |
-> (indexname*typ)list * (indexname*term)list
|
|
21 |
val eta_matches: type_sig -> term * term -> bool
|
|
22 |
val unify: sg * env * (term * term)list -> env
|
|
23 |
exception Unif
|
|
24 |
exception MATCH
|
|
25 |
exception Pattern
|
|
26 |
end;
|
|
27 |
|
|
28 |
functor PatternFun(structure Sign:SIGN and Envir:ENVIR): PATTERN =
|
|
29 |
struct
|
|
30 |
|
|
31 |
structure Type = Sign.Type;
|
|
32 |
|
|
33 |
type type_sig = Type.type_sig
|
|
34 |
type sg = Sign.sg
|
|
35 |
type env = Envir.env
|
|
36 |
|
|
37 |
exception Unif;
|
|
38 |
exception Pattern;
|
|
39 |
|
|
40 |
fun occurs(F,t,env) =
|
|
41 |
let fun occ(Var(G,_)) = (case Envir.lookup(env,G) of
|
|
42 |
Some(t) => occ t
|
|
43 |
| None => F=G)
|
|
44 |
| occ(t1$t2) = occ t1 orelse occ t2
|
|
45 |
| occ(Abs(_,_,t)) = occ t
|
|
46 |
| occ _ = false
|
|
47 |
in occ t end;
|
|
48 |
|
|
49 |
(* Something's wrong *)
|
|
50 |
fun ill_formed s = error ("Ill-formed argument in "^s);
|
|
51 |
|
|
52 |
|
|
53 |
fun mapbnd f =
|
|
54 |
let fun mpb d (Bound(i)) = if i < d then Bound(i) else Bound(f(i-d)+d)
|
|
55 |
| mpb d (Free(c,T)) = Free(c,T)
|
|
56 |
| mpb d (Const(c,T)) = Const(c,T)
|
|
57 |
| mpb d (Var(iname,T)) = Var(iname,T)
|
|
58 |
| mpb d (Abs(s,T,t)) = Abs(s,T,mpb(d+1) t)
|
|
59 |
| mpb d ((u1 $ u2)) = mpb d (u1)$ mpb d (u2)
|
|
60 |
in mpb 0 end;
|
|
61 |
|
|
62 |
fun idx [] j = ~10000
|
|
63 |
| idx(i::is) j = if i=j then length is else idx is j;
|
|
64 |
|
|
65 |
val nth_type = snd o nth_elem;
|
|
66 |
|
|
67 |
fun at xs i = nth_elem (i,xs);
|
|
68 |
|
|
69 |
fun mkabs (binders,is,t) =
|
|
70 |
let fun mk(i::is) = let val (x,T) = nth_elem(i,binders)
|
|
71 |
in Abs(x,T,mk is) end
|
|
72 |
| mk [] = t
|
|
73 |
in mk is end;
|
|
74 |
|
|
75 |
val incr = mapbnd (fn i => i+1);
|
|
76 |
|
|
77 |
(* termlist --> intlist *)
|
|
78 |
fun ints_of [] = []
|
|
79 |
| ints_of (Bound i ::bs) =
|
|
80 |
let val is = ints_of bs
|
|
81 |
in if i mem is then raise Pattern else i::is end
|
|
82 |
| ints_of _ = raise Pattern;
|
|
83 |
|
|
84 |
|
|
85 |
fun app (s,(i::is)) = app (s$Bound(i),is)
|
|
86 |
| app (s,[]) = s;
|
|
87 |
|
|
88 |
fun red (Abs(_,_,s)) (i::is) js = red s is (i::js)
|
|
89 |
| red s is jn = app (mapbnd (at jn) s,is);
|
|
90 |
|
|
91 |
(* split_type ([T1,....,Tn]---> T,n,[]) = ([Tn,...,T1],T) *)
|
|
92 |
fun split_type (T,0,Ts) = (Ts,T)
|
|
93 |
| split_type (Type ("fun",[T1,T2]),n,Ts) = split_type (T2,n-1,T1::Ts)
|
|
94 |
| split_type _ = ill_formed("split_type");
|
|
95 |
|
|
96 |
fun type_of_G (T,n,is) =
|
|
97 |
let val (Ts,U) = split_type(T,n,[]) in map(at Ts)is ---> U end;
|
|
98 |
|
|
99 |
fun mkhnf (binders,is,G,js) = mkabs (binders, is, app(G,js));
|
|
100 |
|
|
101 |
fun mknewhnf(env,binders,is,F as (a,_),T,js) =
|
|
102 |
let val (env',G) = Envir.genvar a (env,type_of_G(T,length is,js))
|
|
103 |
in Envir.update((F,mkhnf(binders,is,G,js)),env') end;
|
|
104 |
|
|
105 |
|
|
106 |
fun devar env t = case strip_comb t of
|
|
107 |
(Var(F,_),ys) =>
|
|
108 |
(case Envir.lookup(env,F) of
|
|
109 |
Some(t) => devar env (red t (ints_of ys) [])
|
|
110 |
| None => t)
|
|
111 |
| _ => t;
|
|
112 |
|
|
113 |
|
|
114 |
(* mk_proj_list(is) = [ |is| - k | 1 <= k <= |is| and is[k] >= 0 ] *)
|
|
115 |
fun mk_proj_list is =
|
|
116 |
let fun mk(i::is,j) = if i >= 0 then j :: mk(is,j-1) else mk(is,j-1)
|
|
117 |
| mk([],_) = []
|
|
118 |
in mk(is,length is - 1) end;
|
|
119 |
|
|
120 |
fun proj(s,env,binders,is) =
|
|
121 |
let fun trans d i = if i<d then i else (idx is (i-d))+d;
|
|
122 |
fun pr(s,env,d,binders) = (case devar env s of
|
|
123 |
Abs(a,T,t) => let val (t',env') = pr(t,env,d+1,((a,T)::binders))
|
|
124 |
in (Abs(a,T,t'),env') end
|
|
125 |
| t => (case strip_comb t of
|
|
126 |
(c as Const _,ts) =>
|
|
127 |
let val (ts',env') = prs(ts,env,d,binders)
|
|
128 |
in (list_comb(c,ts'),env') end
|
|
129 |
| (f as Free _,ts) =>
|
|
130 |
let val (ts',env') = prs(ts,env,d,binders)
|
|
131 |
in (list_comb(f,ts'),env') end
|
|
132 |
| (Bound(i),ts) =>
|
|
133 |
let val j = trans d i
|
|
134 |
in if j < 0 then raise Unif
|
|
135 |
else let val (ts',env') = prs(ts,env,d,binders)
|
|
136 |
in (list_comb(Bound j,ts'),env') end
|
|
137 |
end
|
|
138 |
| (Var(F as (a,_),Fty),ts) =>
|
|
139 |
let val js = ints_of ts;
|
|
140 |
val js' = map (trans d) js;
|
|
141 |
val ks = mk_proj_list js';
|
|
142 |
val ls = filter (fn i => i >= 0) js'
|
|
143 |
val Hty = type_of_G(Fty,length js,ks)
|
|
144 |
val (env',H) = Envir.genvar a (env,Hty)
|
|
145 |
val env'' =
|
|
146 |
Envir.update((F,mkhnf(binders,js,H,ks)),env')
|
|
147 |
in (app(H,ls),env'') end
|
|
148 |
| _ => raise Pattern))
|
|
149 |
and prs(s::ss,env,d,binders) =
|
|
150 |
let val (s',env1) = pr(s,env,d,binders)
|
|
151 |
val (ss',env2) = prs(ss,env1,d,binders)
|
|
152 |
in (s'::ss',env2) end
|
|
153 |
| prs([],env,_,_) = ([],env)
|
|
154 |
in if downto0(is,length binders - 1) then (s,env)
|
|
155 |
else pr(s,env,0,binders)
|
|
156 |
end;
|
|
157 |
|
|
158 |
|
|
159 |
(* mk_ff_list(is,js) = [ length(is) - k | 1 <= k <= |is| and is[k] = js[k] ] *)
|
|
160 |
fun mk_ff_list(is,js) =
|
|
161 |
let fun mk([],[],_) = []
|
|
162 |
| mk(i::is,j::js, k) = if i=j then k :: mk(is,js,k-1)
|
|
163 |
else mk(is,js,k-1)
|
|
164 |
| mk _ = ill_formed"mk_ff_list"
|
|
165 |
in mk(is,js,length is-1) end;
|
|
166 |
|
|
167 |
fun flexflex1(env,binders,F,Fty,is,js) =
|
|
168 |
if is=js then env
|
|
169 |
else let val ks = mk_ff_list(is,js)
|
|
170 |
in mknewhnf(env,binders,is,F,Fty,ks) end;
|
|
171 |
|
|
172 |
fun flexflex2(env,binders,F,Fty,is,G,Gty,js) =
|
|
173 |
let fun ff(F,Fty,is,G as (a,_),Gty,js) =
|
|
174 |
if js subset is
|
|
175 |
then let val t= mkabs(binders,is,app(Var(G,Gty),map (idx is) js))
|
|
176 |
in Envir.update((F,t),env) end
|
|
177 |
else let val ks = is inter js
|
|
178 |
val Hty = type_of_G(Fty,length is,map (idx is) ks)
|
|
179 |
val (env',H) = Envir.genvar a (env,Hty)
|
|
180 |
fun lam(is) = mkabs(binders,is,app(H,map (idx is) ks));
|
|
181 |
in Envir.update((G,lam js), Envir.update((F,lam is),env'))
|
|
182 |
end;
|
|
183 |
in if xless(G,F) then ff(F,Fty,is,G,Gty,js) else ff(G,Gty,js,F,Fty,is) end
|
|
184 |
|
|
185 |
val tsgr = ref(Type.tsig0);
|
|
186 |
|
|
187 |
fun unify_types(T,U, env as Envir.Envir{asol,iTs,maxidx}) =
|
|
188 |
if T=U then env
|
|
189 |
else let val iTs' = Type.unify (!tsgr) ((U,T),iTs)
|
|
190 |
in Envir.Envir{asol=asol,maxidx=maxidx,iTs=iTs'} end
|
|
191 |
handle Type.TUNIFY => raise Unif;
|
|
192 |
|
|
193 |
fun unif binders (env,(s,t)) = case (devar env s,devar env t) of
|
|
194 |
(Abs(ns,Ts,ts),Abs(nt,Tt,tt)) =>
|
|
195 |
let val name = if ns = "" then nt else ns
|
|
196 |
in unif ((name,Ts)::binders) (env,(ts,tt)) end
|
|
197 |
| (Abs(ns,Ts,ts),t) => unif ((ns,Ts)::binders) (env,(ts,(incr t)$Bound(0)))
|
|
198 |
| (t,Abs(nt,Tt,tt)) => unif ((nt,Tt)::binders) (env,((incr t)$Bound(0),tt))
|
|
199 |
| p => cases(binders,env,p)
|
|
200 |
|
|
201 |
and cases(binders,env,(s,t)) = case (strip_comb s,strip_comb t) of
|
|
202 |
((Var(F,Fty),ss),(Var(G,Gty),ts)) =>
|
|
203 |
if F = G then flexflex1(env,binders,F,Fty,ints_of ss,ints_of ts)
|
|
204 |
else flexflex2(env,binders,F,Fty,ints_of ss,G,Gty,ints_of ts)
|
|
205 |
| ((Var(F,_),ss),_) => flexrigid(env,binders,F,ints_of ss,t)
|
|
206 |
| (_,(Var(F,_),ts)) => flexrigid(env,binders,F,ints_of ts,s)
|
|
207 |
| ((Const c,ss),(Const d,ts)) => rigidrigid(env,binders,c,d,ss,ts)
|
|
208 |
| ((Free(f),ss),(Free(g),ts)) => rigidrigid(env,binders,f,g,ss,ts)
|
|
209 |
| ((Bound(i),ss),(Bound(j),ts)) => rigidrigidB (env,binders,i,j,ss,ts)
|
|
210 |
| ((Abs(_),_),_) => raise Pattern
|
|
211 |
| (_,(Abs(_),_)) => raise Pattern
|
|
212 |
| _ => raise Unif
|
|
213 |
|
|
214 |
and rigidrigid (env,binders,(a,Ta),(b,Tb),ss,ts) =
|
|
215 |
if a<>b then raise Unif
|
|
216 |
else foldl (unif binders) (unify_types(Ta,Tb,env), ss~~ts)
|
|
217 |
|
|
218 |
and rigidrigidB (env,binders,i,j,ss,ts) =
|
|
219 |
if i <> j then raise Unif else foldl (unif binders) (env ,ss~~ts)
|
|
220 |
|
|
221 |
and flexrigid (env,binders,F,is,t) =
|
|
222 |
if occurs(F,t,env) then raise Unif
|
|
223 |
else let val (u,env') = proj(t,env,binders,is)
|
|
224 |
in Envir.update((F,mkabs(binders,is,u)),env') end;
|
|
225 |
|
|
226 |
fun unify(sg,env,tus) = (tsgr := #tsig(Sign.rep_sg sg);
|
|
227 |
foldl (unif []) (env,tus));
|
|
228 |
|
|
229 |
|
|
230 |
(*Perform eta-contractions upon a term*)
|
|
231 |
fun eta_contract (Abs(a,T,body)) =
|
|
232 |
(case eta_contract body of
|
|
233 |
body' as (f $ Bound i) =>
|
|
234 |
if i=0 andalso not (0 mem loose_bnos f) then incr_boundvars ~1 f
|
|
235 |
else Abs(a,T,body')
|
|
236 |
| body' => Abs(a,T,body'))
|
|
237 |
| eta_contract(f$t) = eta_contract f $ eta_contract t
|
|
238 |
| eta_contract t = t;
|
|
239 |
|
|
240 |
|
|
241 |
(* Pattern matching. Raises MATCH if non-pattern *)
|
|
242 |
exception MATCH;
|
|
243 |
(* something wron with types, esp in abstractions
|
|
244 |
fun typ_match args = Type.typ_match (!tsgr) args
|
|
245 |
handle Type.TYPE_MATCH => raise MATCH;
|
|
246 |
|
|
247 |
fun match_bind(itms,binders,ixn,is,t) =
|
|
248 |
let val js = loose_bnos t
|
|
249 |
in if null is
|
|
250 |
then if null js then (ixn,t)::itms else raise MATCH
|
|
251 |
else if js subset is
|
|
252 |
then let val t' = if downto0(is,length binders - 1) then t
|
|
253 |
else mapbnd (idx is) t
|
|
254 |
in (ixn, eta_contract(mkabs(binders,is,t'))) :: itms end
|
|
255 |
else raise MATCH
|
|
256 |
end;
|
|
257 |
|
|
258 |
fun match_rr (iTs,(a,Ta),(b,Tb)) =
|
|
259 |
if a<>b then raise MATCH else typ_match (iTs,(Ta,Tb))
|
|
260 |
|
|
261 |
(* Pre: pat and obj have same type *)
|
|
262 |
fun mtch(binders,env as (iTs,itms),pat,obj) = case pat of
|
|
263 |
Var(ixn,_) => (case assoc(itms,ixn) of
|
|
264 |
None => (iTs,match_bind(itms,binders,ixn,[],obj))
|
|
265 |
| Some u => if obj aconv u then env else raise MATCH)
|
|
266 |
| Abs(ns,Ts,ts) =>
|
|
267 |
(case obj of
|
|
268 |
Abs(nt,Tt,tt) => mtch((nt,Tt)::binders,env,ts,tt)
|
|
269 |
| _ => let val Tt = typ_subst_TVars iTs Ts
|
|
270 |
in mtch((ns,Tt)::binders,env,ts,(incr obj)$Bound(0)) end)
|
|
271 |
| _ => (case obj of
|
|
272 |
Abs(nt,Tt,tt) =>
|
|
273 |
mtch((nt,Tt)::binders,env,(incr pat)$Bound(0),tt)
|
|
274 |
| _ => cases(binders,env,pat,obj))
|
|
275 |
|
|
276 |
and cases(binders,env as (iTs,itms),pat,obj) =
|
|
277 |
let fun structural() = case (pat,obj) of
|
|
278 |
(Const c,Const d) => (match_rr(iTs,c,d),itms)
|
|
279 |
| (Free f,Free g) => (match_rr(iTs,f,g),itms)
|
|
280 |
| (Bound i,Bound j) => if i=j then env else raise MATCH
|
|
281 |
| (f$t,g$u) => mtch(binders,mtch(binders,env,t,u),f,g)
|
|
282 |
| _ => raise MATCH
|
|
283 |
in case strip_comb pat of
|
|
284 |
(Var(ixn,_),bs) =>
|
|
285 |
(let val is = ints_of bs
|
|
286 |
in case assoc(itms,ixn) of
|
|
287 |
None => (iTs,match_bind(itms,binders,ixn,is,obj))
|
|
288 |
| Some u => if obj aconv (red u is []) then env else raise MATCH
|
|
289 |
end (* if ints_of fails: *) handle Pattern => structural())
|
|
290 |
| _ => structural()
|
|
291 |
end;
|
|
292 |
|
|
293 |
fun match tsg = (tsgr := tsg;
|
|
294 |
fn (pat,obj) =>
|
63
|
295 |
let val pT = fastype_of pat
|
|
296 |
and oT = fastype_of obj
|
|
297 |
val iTs = typ_match ([],(pT,oT))
|
0
|
298 |
in mtch([], (iTs,[]), pat, eta_contract obj)
|
|
299 |
handle Pattern => raise MATCH
|
|
300 |
end)
|
|
301 |
|
|
302 |
(*Predicate: does the pattern match the object?*)
|
|
303 |
fun matches tsig args = (match tsig args; true)
|
|
304 |
handle MATCH => false;
|
|
305 |
*)
|
|
306 |
|
|
307 |
(*First-order matching; term_match tsig (pattern, object)
|
|
308 |
returns a (tyvar,typ)list and (var,term)list.
|
|
309 |
The pattern and object may have variables in common.
|
|
310 |
Instantiation does not affect the object, so matching ?a with ?a+1 works.
|
|
311 |
A Const does not match a Free of the same name!
|
|
312 |
Does not notice eta-equality, thus f does not match %(x)f(x) *)
|
|
313 |
fun match tsig (pat,obj) =
|
|
314 |
let fun typ_match args = (Type.typ_match tsig args)
|
|
315 |
handle Type.TYPE_MATCH => raise MATCH;
|
|
316 |
fun mtch (tyinsts,insts) = fn
|
|
317 |
(Var(ixn,T), t) =>
|
|
318 |
if null (loose_bnos t)
|
|
319 |
then case assoc(insts,ixn) of
|
63
|
320 |
None => (typ_match (tyinsts, (T, fastype_of t)),
|
0
|
321 |
(ixn,t)::insts)
|
|
322 |
| Some u => if t aconv u then (tyinsts,insts) else raise MATCH
|
|
323 |
else raise MATCH
|
|
324 |
| (Free (a,T), Free (b,U)) =>
|
|
325 |
if a=b then (typ_match (tyinsts,(T,U)), insts) else raise MATCH
|
|
326 |
| (Const (a,T), Const (b,U)) =>
|
|
327 |
if a=b then (typ_match (tyinsts,(T,U)), insts) else raise MATCH
|
|
328 |
| (Bound i, Bound j) =>
|
|
329 |
if i=j then (tyinsts,insts) else raise MATCH
|
|
330 |
| (Abs(_,T,t), Abs(_,U,u)) =>
|
|
331 |
mtch (typ_match (tyinsts,(T,U)),insts) (t,u)
|
|
332 |
| (f$t, g$u) => mtch (mtch (tyinsts,insts) (f,g)) (t, u)
|
|
333 |
| _ => raise MATCH
|
|
334 |
in mtch([],[]) (pat,obj) end;
|
|
335 |
|
|
336 |
(*Predicate: does the pattern match the object?*)
|
|
337 |
fun eta_matches tsig (pat,obj) =
|
|
338 |
(match tsig (eta_contract pat,eta_contract obj); true)
|
|
339 |
handle MATCH => false;
|
|
340 |
|
|
341 |
end;
|