(* Title: pattern
ID: $Id$
Author: Tobias Nipkow and Christine Heinzelmann, TU Muenchen
Copyright 1993 TU Muenchen
Unification of Higher-Order Patterns.
See also:
Tobias Nipkow. Functional Unification of Higher-Order Patterns.
In Proceedings of the 8th IEEE Symposium Logic in Computer Science, 1993.
TODO: optimize red by special-casing it
*)
signature PATTERN =
sig
type type_sig
type sg
type env
val eta_contract: term -> term
val match: type_sig -> term * term
-> (indexname*typ)list * (indexname*term)list
val matches: type_sig -> term * term -> bool
val unify: sg * env * (term * term)list -> env
exception Unif
exception MATCH
exception Pattern
end;
functor PatternFun(structure Sign:SIGN and Envir:ENVIR): PATTERN =
struct
structure Type = Sign.Type;
type type_sig = Type.type_sig
type sg = Sign.sg
type env = Envir.env
exception Unif;
exception Pattern;
fun occurs(F,t,env) =
let fun occ(Var(G,_)) = (case Envir.lookup(env,G) of
Some(t) => occ t
| None => F=G)
| occ(t1$t2) = occ t1 orelse occ t2
| occ(Abs(_,_,t)) = occ t
| occ _ = false
in occ t end;
fun mapbnd f =
let fun mpb d (Bound(i)) = if i < d then Bound(i) else Bound(f(i-d)+d)
| mpb d (Abs(s,T,t)) = Abs(s,T,mpb(d+1) t)
| mpb d ((u1 $ u2)) = (mpb d u1)$(mpb d u2)
| mpb _ atom = atom
in mpb 0 end;
fun idx [] j = ~10000
| idx(i::is) j = if i=j then length is else idx is j;
val nth_type = snd o nth_elem;
fun at xs i = nth_elem (i,xs);
fun mkabs (binders,is,t) =
let fun mk(i::is) = let val (x,T) = nth_elem(i,binders)
in Abs(x,T,mk is) end
| mk [] = t
in mk is end;
val incr = mapbnd (fn i => i+1);
fun ints_of [] = []
| ints_of (Bound i ::bs) =
let val is = ints_of bs
in if i mem is then raise Pattern else i::is end
| ints_of _ = raise Pattern;
fun app (s,(i::is)) = app (s$Bound(i),is)
| app (s,[]) = s;
fun red (Abs(_,_,s)) (i::is) js = red s is (i::js)
| red t [] [] = t
| red t is jn = app (mapbnd (at jn) t,is);
(* split_type ([T1,....,Tn]---> T,n,[]) = ([Tn,...,T1],T) *)
fun split_type (T,0,Ts) = (Ts,T)
| split_type (Type ("fun",[T1,T2]),n,Ts) = split_type (T2,n-1,T1::Ts)
| split_type _ = error("split_type");
fun type_of_G (T,n,is) =
let val (Ts,U) = split_type(T,n,[]) in map(at Ts)is ---> U end;
fun mkhnf (binders,is,G,js) = mkabs (binders, is, app(G,js));
fun mknewhnf(env,binders,is,F as (a,_),T,js) =
let val (env',G) = Envir.genvar a (env,type_of_G(T,length is,js))
in Envir.update((F,mkhnf(binders,is,G,js)),env') end;
fun devar env t = case strip_comb t of
(Var(F,_),ys) =>
(case Envir.lookup(env,F) of
Some(t) => devar env (red t (ints_of ys) [])
| None => t)
| _ => t;
(* mk_proj_list(is) = [ |is| - k | 1 <= k <= |is| and is[k] >= 0 ] *)
fun mk_proj_list is =
let fun mk(i::is,j) = if i >= 0 then j :: mk(is,j-1) else mk(is,j-1)
| mk([],_) = []
in mk(is,length is - 1) end;
fun proj(s,env,binders,is) =
let fun trans d i = if i<d then i else (idx is (i-d))+d;
fun pr(s,env,d,binders) = (case devar env s of
Abs(a,T,t) => let val (t',env') = pr(t,env,d+1,((a,T)::binders))
in (Abs(a,T,t'),env') end
| t => (case strip_comb t of
(c as Const _,ts) =>
let val (ts',env') = prs(ts,env,d,binders)
in (list_comb(c,ts'),env') end
| (f as Free _,ts) =>
let val (ts',env') = prs(ts,env,d,binders)
in (list_comb(f,ts'),env') end
| (Bound(i),ts) =>
let val j = trans d i
in if j < 0 then raise Unif
else let val (ts',env') = prs(ts,env,d,binders)
in (list_comb(Bound j,ts'),env') end
end
| (Var(F as (a,_),Fty),ts) =>
let val js = ints_of ts;
val js' = map (trans d) js;
val ks = mk_proj_list js';
val ls = filter (fn i => i >= 0) js'
val Hty = type_of_G(Fty,length js,ks)
val (env',H) = Envir.genvar a (env,Hty)
val env'' =
Envir.update((F,mkhnf(binders,js,H,ks)),env')
in (app(H,ls),env'') end
| _ => raise Pattern))
and prs(s::ss,env,d,binders) =
let val (s',env1) = pr(s,env,d,binders)
val (ss',env2) = prs(ss,env1,d,binders)
in (s'::ss',env2) end
| prs([],env,_,_) = ([],env)
in if downto0(is,length binders - 1) then (s,env)
else pr(s,env,0,binders)
end;
(* mk_ff_list(is,js) = [ length(is) - k | 1 <= k <= |is| and is[k] = js[k] ] *)
fun mk_ff_list(is,js) =
let fun mk([],[],_) = []
| mk(i::is,j::js, k) = if i=j then k :: mk(is,js,k-1)
else mk(is,js,k-1)
| mk _ = error"mk_ff_list"
in mk(is,js,length is-1) end;
fun flexflex1(env,binders,F,Fty,is,js) =
if is=js then env
else let val ks = mk_ff_list(is,js)
in mknewhnf(env,binders,is,F,Fty,ks) end;
fun flexflex2(env,binders,F,Fty,is,G,Gty,js) =
let fun ff(F,Fty,is,G as (a,_),Gty,js) =
if js subset is
then let val t= mkabs(binders,is,app(Var(G,Gty),map (idx is) js))
in Envir.update((F,t),env) end
else let val ks = is inter js
val Hty = type_of_G(Fty,length is,map (idx is) ks)
val (env',H) = Envir.genvar a (env,Hty)
fun lam(is) = mkabs(binders,is,app(H,map (idx is) ks));
in Envir.update((G,lam js), Envir.update((F,lam is),env'))
end;
in if xless(G,F) then ff(F,Fty,is,G,Gty,js) else ff(G,Gty,js,F,Fty,is) end
val tsgr = ref(Type.tsig0);
fun unify_types(T,U, env as Envir.Envir{asol,iTs,maxidx}) =
if T=U then env
else let val (iTs',maxidx') = Type.unify (!tsgr) maxidx iTs (U,T)
in Envir.Envir{asol=asol,maxidx=maxidx',iTs=iTs'} end
handle Type.TUNIFY => raise Unif;
fun unif binders (env,(s,t)) = case (devar env s,devar env t) of
(Abs(ns,Ts,ts),Abs(nt,Tt,tt)) =>
let val name = if ns = "" then nt else ns
in unif ((name,Ts)::binders) (env,(ts,tt)) end
| (Abs(ns,Ts,ts),t) => unif ((ns,Ts)::binders) (env,(ts,(incr t)$Bound(0)))
| (t,Abs(nt,Tt,tt)) => unif ((nt,Tt)::binders) (env,((incr t)$Bound(0),tt))
| p => cases(binders,env,p)
and cases(binders,env,(s,t)) = case (strip_comb s,strip_comb t) of
((Var(F,Fty),ss),(Var(G,Gty),ts)) =>
if F = G then flexflex1(env,binders,F,Fty,ints_of ss,ints_of ts)
else flexflex2(env,binders,F,Fty,ints_of ss,G,Gty,ints_of ts)
| ((Var(F,_),ss),_) => flexrigid(env,binders,F,ints_of ss,t)
| (_,(Var(F,_),ts)) => flexrigid(env,binders,F,ints_of ts,s)
| ((Const c,ss),(Const d,ts)) => rigidrigid(env,binders,c,d,ss,ts)
| ((Free(f),ss),(Free(g),ts)) => rigidrigid(env,binders,f,g,ss,ts)
| ((Bound(i),ss),(Bound(j),ts)) => rigidrigidB (env,binders,i,j,ss,ts)
| ((Abs(_),_),_) => raise Pattern
| (_,(Abs(_),_)) => raise Pattern
| _ => raise Unif
and rigidrigid (env,binders,(a,Ta),(b,Tb),ss,ts) =
if a<>b then raise Unif
else foldl (unif binders) (unify_types(Ta,Tb,env), ss~~ts)
and rigidrigidB (env,binders,i,j,ss,ts) =
if i <> j then raise Unif else foldl (unif binders) (env ,ss~~ts)
and flexrigid (env,binders,F,is,t) =
if occurs(F,t,env) then raise Unif
else let val (u,env') = proj(t,env,binders,is)
in Envir.update((F,mkabs(binders,is,u)),env') end;
fun unify(sg,env,tus) = (tsgr := #tsig(Sign.rep_sg sg);
foldl (unif []) (env,tus));
(*Perform eta-contractions upon a term*)
fun eta_contract (Abs(a,T,body)) =
(case eta_contract body of
body' as (f $ Bound i) =>
if i=0 andalso not (0 mem loose_bnos f) then incr_boundvars ~1 f
else Abs(a,T,body')
| body' => Abs(a,T,body'))
| eta_contract(f$t) = eta_contract f $ eta_contract t
| eta_contract t = t;
(* Pattern matching *)
exception MATCH;
(*First-order matching; term_match tsig (pattern, object)
returns a (tyvar,typ)list and (var,term)list.
The pattern and object may have variables in common.
Instantiation does not affect the object, so matching ?a with ?a+1 works.
A Const does not match a Free of the same name!
Does not notice eta-equality, thus f does not match %(x)f(x) *)
fun fomatch tsig (pat,obj) =
let fun typ_match args = (Type.typ_match tsig args)
handle Type.TYPE_MATCH => raise MATCH;
fun mtch (tyinsts,insts) = fn
(Var(ixn,T), t) =>
if loose_bvar(t,0) then raise MATCH
else (case assoc(insts,ixn) of
None => (typ_match (tyinsts, (T, fastype_of t)),
(ixn,t)::insts)
| Some u => if t aconv u then (tyinsts,insts) else raise MATCH)
| (Free (a,T), Free (b,U)) =>
if a=b then (typ_match (tyinsts,(T,U)), insts) else raise MATCH
| (Const (a,T), Const (b,U)) =>
if a=b then (typ_match (tyinsts,(T,U)), insts) else raise MATCH
| (Bound i, Bound j) =>
if i=j then (tyinsts,insts) else raise MATCH
| (Abs(_,T,t), Abs(_,U,u)) =>
mtch (typ_match (tyinsts,(T,U)),insts) (t,u)
| (f$t, g$u) => mtch (mtch (tyinsts,insts) (f,g)) (t, u)
| _ => raise MATCH
in mtch([],[]) (eta_contract pat,eta_contract obj) end;
fun match_bind(itms,binders,ixn,is,t) =
let val js = loose_bnos t
in if null is
then if null js then (ixn,t)::itms else raise MATCH
else if js subset is
then let val t' = if downto0(is,length binders - 1) then t
else mapbnd (idx is) t
in (ixn, mkabs(binders,is,t')) :: itms end
else raise MATCH
end;
(*Tests whether 2 terms are alpha/eta-convertible and have same type.
Note that Consts and Vars may have more than one type.*)
infix aeconv;
fun (Const(a,T)) aeconv (Const(b,U)) = a=b andalso T=U
| (Free(a,T)) aeconv (Free(b,U)) = a=b andalso T=U
| (Var(v,T)) aeconv (Var(w,U)) = v=w andalso T=U
| (Bound i) aeconv (Bound j) = i=j
| (Abs(_,T,t)) aeconv (Abs(_,U,u)) = (t aeconv u) andalso T=U
| (Abs(_,T,t)) aeconv u = t aeconv ((incr u)$Bound(0))
| t aeconv (Abs(_,U,u)) = ((incr t)$Bound(0)) aeconv u
| (f$t) aeconv (g$u) = (f aeconv g) andalso (t aeconv u)
| _ aeconv _ = false;
fun match tsg (po as (pat,obj)) =
let
fun typ_match args = Type.typ_match tsg args
handle Type.TYPE_MATCH => raise MATCH;
(* Pre: pat and obj have same type *)
fun mtch binders (env as (iTs,itms),(pat,obj)) =
case pat of
Abs(ns,Ts,ts) =>
(case obj of
Abs(nt,Tt,tt) => mtch ((nt,Tt)::binders) (env,(ts,tt))
| _ => let val Tt = typ_subst_TVars iTs Ts
in mtch((ns,Tt)::binders)(env,(ts,(incr obj)$Bound(0))) end)
| _ => (case obj of
Abs(nt,Tt,tt) =>
mtch((nt,Tt)::binders)(env,((incr pat)$Bound(0),tt))
| _ => cases(binders,env,pat,obj))
and cases(binders,env as (iTs,itms),pat,obj) =
let val (ph,pargs) = strip_comb pat
fun rigrig1(iTs,oargs) =
foldl (mtch binders) ((iTs,itms), pargs~~oargs)
fun rigrig2((a,Ta),(b,Tb),oargs) =
if a<> b then raise MATCH
else rigrig1(typ_match(iTs,(Ta,Tb)), oargs)
in case ph of
Var(ixn,_) =>
let val is = ints_of pargs
in case assoc(itms,ixn) of
None => (iTs,match_bind(itms,binders,ixn,is,obj))
| Some u => if obj aeconv (red u is []) then env
else raise MATCH
end
| _ =>
let val (oh,oargs) = strip_comb obj
in case (ph,oh) of
(Const c,Const d) => rigrig2(c,d,oargs)
| (Free f,Free g) => rigrig2(f,g,oargs)
| (Bound i,Bound j) => if i<>j then raise MATCH
else rigrig1(iTs,oargs)
| (Abs _, _) => raise Pattern
| (_, Abs _) => raise Pattern
| _ => raise MATCH
end
end;
val pT = fastype_of pat
and oT = fastype_of obj
val iTs = typ_match ([],(pT,oT))
in mtch [] ((iTs,[]), po)
handle Pattern => fomatch tsg po
end;
(*Predicate: does the pattern match the object?*)
fun matches tsig po = (match tsig po; true) handle MATCH => false;
end;