(* Title: Pure/pattern.ML
ID: $Id$
Author: Tobias Nipkow, Christine Heinzelmann, and Stefan Berghofer
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
*)
infix aeconv;
signature PATTERN =
sig
val trace_unify_fail: bool ref
val aeconv: term * term -> bool
val eta_contract: term -> term
val eta_long: typ list -> term -> term
val beta_eta_contract: term -> term
val eta_contract_atom: term -> term
val match: theory -> term * term -> Type.tyenv * Envir.tenv
val first_order_match: theory -> term * term -> Type.tyenv * Envir.tenv
val matches: theory -> term * term -> bool
val matches_subterm: theory -> term * term -> bool
val unify: theory * Envir.env * (term * term)list -> Envir.env
val first_order: term -> bool
val pattern: term -> bool
val rewrite_term: theory -> (term * term) list -> (term -> term option) list -> term -> term
exception Unif
exception MATCH
exception Pattern
end;
structure Pattern: PATTERN =
struct
exception Unif;
exception Pattern;
val trace_unify_fail = ref false;
fun string_of_term thy env binders t = Sign.string_of_term thy
(Envir.norm_term env (subst_bounds(map Free binders,t)));
fun bname binders i = fst(List.nth(binders,i));
fun bnames binders is = space_implode " " (map (bname binders) is);
fun typ_clash thy (tye,T,U) =
if !trace_unify_fail
then let val t = Sign.string_of_typ thy (Envir.norm_type tye T)
and u = Sign.string_of_typ thy (Envir.norm_type tye U)
in tracing("The following types do not unify:\n" ^ t ^ "\n" ^ u) end
else ()
fun clash a b =
if !trace_unify_fail then tracing("Clash: " ^ a ^ " =/= " ^ b) else ()
fun boundVar binders i =
"bound variable " ^ bname binders i ^ " (depth " ^ string_of_int i ^ ")";
fun clashBB binders i j =
if !trace_unify_fail then clash (boundVar binders i) (boundVar binders j)
else ()
fun clashB binders i s =
if !trace_unify_fail then clash (boundVar binders i) s
else ()
fun proj_fail thy (env,binders,F,_,is,t) =
if !trace_unify_fail
then let val f = Syntax.string_of_vname F
val xs = bnames binders is
val u = string_of_term thy env binders t
val ys = bnames binders (loose_bnos t \\ is)
in tracing("Cannot unify variable " ^ f ^
" (depending on bound variables " ^ xs ^ ")\nwith term " ^ u ^
"\nTerm contains additional bound variable(s) " ^ ys)
end
else ()
fun ocheck_fail thy (F,t,binders,env) =
if !trace_unify_fail
then let val f = Syntax.string_of_vname F
val u = string_of_term thy env binders t
in tracing("Variable " ^ f ^ " occurs in term\n" ^ u ^
"\nCannot unify!\n")
end
else ()
fun occurs(F,t,env) =
let fun occ(Var (G, T)) = (case Envir.lookup (env, (G, T)) 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)$(mpb d u2)
| mpb _ atom = atom
in mpb 0 end;
fun idx [] j = raise Unif
| idx(i::is) j = if (i:int) =j then length is else idx is j;
fun at xs i = List.nth (xs,i);
fun mkabs (binders,is,t) =
let fun mk(i::is) = let val (x,T) = List.nth(binders,i)
in Abs(x,T,mk is) end
| mk [] = t
in mk is end;
val incr = mapbnd (fn i => i+1);
fun ints_of [] = []
| ints_of (Bound i ::bs) =
let val is = ints_of bs
in if i mem_int is then raise Pattern else i::is end
| ints_of _ = raise Pattern;
fun ints_of' env ts = ints_of (map (Envir.head_norm env) ts);
fun app (s,(i::is)) = app (s$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 (env as Envir.Envir {iTs, ...}) (T,n,is) =
let val (Ts, U) = split_type (Envir.norm_type iTs 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 env (T,length is,js))
in Envir.update (((F, T), mkhnf (binders, is, G, js)), env') end;
(* 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 isSome i 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 Envir.head_norm 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
val (ts',env') = prs(ts,env,d,binders)
in (list_comb(Bound j,ts'),env') end
| (Var(F as (a,_),Fty),ts) =>
let val js = ints_of' env ts;
val js' = map (try (trans d)) js;
val ks = mk_proj_list js';
val ls = List.mapPartial I js'
val Hty = type_of_G env (Fty,length js,ks)
val (env',H) = Envir.genvar a (env,Hty)
val env'' =
Envir.update (((F, Fty), 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:int) = 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_int is
then let val t= mkabs(binders,is,app(Var(G,Gty),map (idx is) js))
in Envir.update (((F, Fty), t), env) end
else let val ks = is inter_int js
val Hty = type_of_G env (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, Gty), lam js), Envir.update (((F, Fty), 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
fun unify_types thy (T,U, env as Envir.Envir{asol,iTs,maxidx}) =
if T=U then env
else let val (iTs',maxidx') = Sign.typ_unify thy (U, T) (iTs, maxidx)
in Envir.Envir{asol=asol,maxidx=maxidx',iTs=iTs'} end
handle Type.TUNIFY => (typ_clash thy (iTs,T,U); raise Unif);
fun unif thy binders (env,(s,t)) = case (Envir.head_norm env s, Envir.head_norm env t) of
(Abs(ns,Ts,ts),Abs(nt,Tt,tt)) =>
let val name = if ns = "" then nt else ns
in unif thy ((name,Ts)::binders) (env,(ts,tt)) end
| (Abs(ns,Ts,ts),t) => unif thy ((ns,Ts)::binders) (env,(ts,(incr t)$Bound(0)))
| (t,Abs(nt,Tt,tt)) => unif thy ((nt,Tt)::binders) (env,((incr t)$Bound(0),tt))
| p => cases thy (binders,env,p)
and cases thy (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' env ss,ints_of' env ts)
else flexflex2(env,binders,F,Fty,ints_of' env ss,G,Gty,ints_of' env ts)
| ((Var(F,Fty),ss),_) => flexrigid thy (env,binders,F,Fty,ints_of' env ss,t)
| (_,(Var(F,Fty),ts)) => flexrigid thy (env,binders,F,Fty,ints_of' env ts,s)
| ((Const c,ss),(Const d,ts)) => rigidrigid thy (env,binders,c,d,ss,ts)
| ((Free(f),ss),(Free(g),ts)) => rigidrigid thy (env,binders,f,g,ss,ts)
| ((Bound(i),ss),(Bound(j),ts)) => rigidrigidB thy (env,binders,i,j,ss,ts)
| ((Abs(_),_),_) => raise Pattern
| (_,(Abs(_),_)) => raise Pattern
| ((Const(c,_),_),(Free(f,_),_)) => (clash c f; raise Unif)
| ((Const(c,_),_),(Bound i,_)) => (clashB binders i c; raise Unif)
| ((Free(f,_),_),(Const(c,_),_)) => (clash f c; raise Unif)
| ((Free(f,_),_),(Bound i,_)) => (clashB binders i f; raise Unif)
| ((Bound i,_),(Const(c,_),_)) => (clashB binders i c; raise Unif)
| ((Bound i,_),(Free(f,_),_)) => (clashB binders i f; raise Unif)
and rigidrigid thy (env,binders,(a,Ta),(b,Tb),ss,ts) =
if a<>b then (clash a b; raise Unif)
else Library.foldl (unif thy binders) (unify_types thy (Ta,Tb,env), ss~~ts)
and rigidrigidB thy (env,binders,i,j,ss,ts) =
if i <> j then (clashBB binders i j; raise Unif)
else Library.foldl (unif thy binders) (env ,ss~~ts)
and flexrigid thy (params as (env,binders,F,Fty,is,t)) =
if occurs(F,t,env) then (ocheck_fail thy (F,t,binders,env); raise Unif)
else (let val (u,env') = proj(t,env,binders,is)
in Envir.update (((F, Fty), mkabs (binders, is, u)), env') end
handle Unif => (proj_fail thy params; raise Unif));
fun unify(thy,env,tus) = Library.foldl (unif thy []) (env,tus);
(*Eta-contract a term (fully)*)
fun eta_contract t =
let
exception SAME;
fun eta (Abs (a, T, body)) =
((case eta body of
body' as (f $ Bound 0) =>
if loose_bvar1 (f, 0) then Abs(a, T, body')
else incr_boundvars ~1 f
| body' => Abs (a, T, body')) handle SAME =>
(case body of
(f $ Bound 0) =>
if loose_bvar1 (f, 0) then raise SAME
else incr_boundvars ~1 f
| _ => raise SAME))
| eta (f $ t) =
(let val f' = eta f
in f' $ etah t end handle SAME => f $ eta t)
| eta _ = raise SAME
and etah t = (eta t handle SAME => t)
in etah t end;
val beta_eta_contract = eta_contract o Envir.beta_norm;
(*Eta-contract a term from outside: just enough to reduce it to an atom
DOESN'T QUITE WORK!
*)
fun eta_contract_atom (t0 as Abs(a, T, body)) =
(case eta_contract2 body of
body' as (f $ Bound 0) =>
if loose_bvar1(f,0) then Abs(a,T,body')
else eta_contract_atom (incr_boundvars ~1 f)
| _ => t0)
| eta_contract_atom t = t
and eta_contract2 (f$t) = f $ eta_contract_atom t
| eta_contract2 t = eta_contract_atom t;
(* put a term into eta long beta normal form *)
fun eta_long Ts (Abs (s, T, t)) = Abs (s, T, eta_long (T :: Ts) t)
| eta_long Ts t = (case strip_comb t of
(Abs _, _) => eta_long Ts (Envir.beta_norm t)
| (u, ts) =>
let
val Us = binder_types (fastype_of1 (Ts, t));
val i = length Us
in list_abs (map (pair "x") Us,
list_comb (incr_boundvars i u, map (eta_long (rev Us @ Ts))
(map (incr_boundvars i) ts @ map Bound (i - 1 downto 0))))
end);
(*Tests whether 2 terms are alpha/eta-convertible and have same type.
Note that Consts and Vars may have more than one type.*)
fun t aeconv u = aconv_aux (eta_contract_atom t, eta_contract_atom u)
and aconv_aux (Const(a,T), Const(b,U)) = a=b andalso T=U
| aconv_aux (Free(a,T), Free(b,U)) = a=b andalso T=U
| aconv_aux (Var(v,T), Var(w,U)) = eq_ix(v,w) andalso T=U
| aconv_aux (Bound i, Bound j) = i=j
| aconv_aux (Abs(_,T,t), Abs(_,U,u)) = (t aeconv u) andalso T=U
| aconv_aux (f$t, g$u) = (f aeconv g) andalso (t aeconv u)
| aconv_aux _ = false;
(*** Matching ***)
exception MATCH;
fun typ_match thy (tyenv, TU) = Sign.typ_match thy TU tyenv
handle Type.TYPE_MATCH => raise MATCH;
(*First-order matching;
fomatch thy (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.
Object is eta-contracted on the fly (by eta-expanding the pattern).
Precondition: the pattern is already eta-contracted!
Note: types are matched on the fly *)
fun fomatch thy =
let
fun mtch (instsp as (tyinsts,insts)) = fn
(Var(ixn,T), t) =>
if loose_bvar(t,0) then raise MATCH
else (case Envir.lookup' (insts, (ixn, T)) of
NONE => (typ_match thy (tyinsts, (T, fastype_of t)),
Vartab.update_new (ixn, (T, t)) insts)
| SOME u => if t aeconv u then instsp else raise MATCH)
| (Free (a,T), Free (b,U)) =>
if a=b then (typ_match thy (tyinsts,(T,U)), insts) else raise MATCH
| (Const (a,T), Const (b,U)) =>
if a=b then (typ_match thy (tyinsts,(T,U)), insts) else raise MATCH
| (Bound i, Bound j) => if i=j then instsp else raise MATCH
| (Abs(_,T,t), Abs(_,U,u)) =>
mtch (typ_match thy (tyinsts,(T,U)),insts) (t,u)
| (f$t, g$u) => mtch (mtch instsp (f,g)) (t, u)
| (t, Abs(_,U,u)) => mtch instsp ((incr t)$(Bound 0), u)
| _ => raise MATCH
in mtch end;
fun first_order_match thy = fomatch thy (Vartab.empty, Vartab.empty);
(* Matching of higher-order patterns *)
fun match_bind(itms,binders,ixn,T,is,t) =
let val js = loose_bnos t
in if null is
then if null js then Vartab.update_new (ixn, (T, t)) itms else raise MATCH
else if js subset_int is
then let val t' = if downto0(is,length binders - 1) then t
else mapbnd (idx is) t
in Vartab.update_new (ixn, (T, mkabs (binders, is, t'))) itms end
else raise MATCH
end;
fun match thy (po as (pat,obj)) =
let
(* Pre: pat and obj have same type *)
fun mtch binders (env as (iTs,itms),(pat,obj)) =
case pat of
Abs(ns,Ts,ts) =>
(case obj of
Abs(nt,Tt,tt) => mtch ((nt,Tt)::binders) (env,(ts,tt))
| _ => let val Tt = Envir.typ_subst_TVars iTs Ts
in mtch((ns,Tt)::binders)(env,(ts,(incr obj)$Bound(0))) end)
| _ => (case obj of
Abs(nt,Tt,tt) =>
mtch((nt,Tt)::binders)(env,((incr pat)$Bound(0),tt))
| _ => cases(binders,env,pat,obj))
and cases(binders,env as (iTs,itms),pat,obj) =
let val (ph,pargs) = strip_comb pat
fun rigrig1(iTs,oargs) =
Library.foldl (mtch binders) ((iTs,itms), pargs~~oargs)
fun rigrig2((a:string,Ta),(b,Tb),oargs) =
if a <> b then raise MATCH
else rigrig1(typ_match thy (iTs,(Ta,Tb)), oargs)
in case ph of
Var(ixn,T) =>
let val is = ints_of pargs
in case Envir.lookup' (itms, (ixn, T)) of
NONE => (iTs,match_bind(itms,binders,ixn,T,is,obj))
| SOME u => if obj aeconv (red u is []) then env
else raise MATCH
end
| _ =>
let val (oh,oargs) = strip_comb obj
in case (ph,oh) of
(Const c,Const d) => rigrig2(c,d,oargs)
| (Free f,Free g) => rigrig2(f,g,oargs)
| (Bound i,Bound j) => if i<>j then raise MATCH
else rigrig1(iTs,oargs)
| (Abs _, _) => raise Pattern
| (_, Abs _) => raise Pattern
| _ => raise MATCH
end
end;
val pT = fastype_of pat
and oT = fastype_of obj
val iTs = typ_match thy (Vartab.empty, (pT,oT))
val insts2 = (iTs, Vartab.empty)
in mtch [] (insts2, po)
handle Pattern => fomatch thy insts2 po
end;
(*Predicate: does the pattern match the object?*)
fun matches thy po = (match thy po; true) handle MATCH => false;
(* Does pat match a subterm of obj? *)
fun matches_subterm thy (pat,obj) =
let fun msub(bounds,obj) = matches thy (pat,obj) orelse
case obj of
Abs(x,T,t) => let val y = variant bounds x
val f = Free(":" ^ y,T)
in msub(x::bounds,subst_bound(f,t)) end
| s$t => msub(bounds,s) orelse msub(bounds,t)
| _ => false
in msub([],obj) end;
fun first_order(Abs(_,_,t)) = first_order t
| first_order(t $ u) = first_order t andalso first_order u andalso
not(is_Var t)
| first_order _ = true;
fun pattern(Abs(_,_,t)) = pattern t
| pattern(t) = let val (head,args) = strip_comb t
in if is_Var head
then let val _ = ints_of args in true end
handle Pattern => false
else forall pattern args
end;
(* rewriting -- simple but fast *)
fun rewrite_term thy rules procs tm =
let
val skel0 = Bound 0;
fun variant_absfree bounds (x, T, t) =
let
val (x', t') = Term.dest_abs (Term.bound bounds, T, t);
fun abs u = Abs (x, T, abstract_over (Free (x', T), u));
in (abs, t') end;
fun match_rew tm (tm1, tm2) =
let val rtm = if_none (Term.rename_abs tm1 tm tm2) tm2 in
SOME (Envir.subst_vars (match thy (tm1, tm)) rtm, rtm)
handle MATCH => NONE
end;
fun rew (Abs (_, _, body) $ t) = SOME (subst_bound (t, body), skel0)
| rew tm = (case get_first (match_rew tm) rules of
NONE => Option.map (rpair skel0) (get_first (fn p => p tm) procs)
| x => x);
fun rew1 bounds (Var _) _ = NONE
| rew1 bounds skel tm = (case rew2 bounds skel tm of
SOME tm1 => (case rew tm1 of
SOME (tm2, skel') => SOME (if_none (rew1 bounds skel' tm2) tm2)
| NONE => SOME tm1)
| NONE => (case rew tm of
SOME (tm1, skel') => SOME (if_none (rew1 bounds skel' tm1) tm1)
| NONE => NONE))
and rew2 bounds skel (tm1 $ tm2) = (case tm1 of
Abs (_, _, body) =>
let val tm' = subst_bound (tm2, body)
in SOME (if_none (rew2 bounds skel0 tm') tm') end
| _ =>
let val (skel1, skel2) = (case skel of
skel1 $ skel2 => (skel1, skel2)
| _ => (skel0, skel0))
in case rew1 bounds skel1 tm1 of
SOME tm1' => (case rew1 bounds skel2 tm2 of
SOME tm2' => SOME (tm1' $ tm2')
| NONE => SOME (tm1' $ tm2))
| NONE => (case rew1 bounds skel2 tm2 of
SOME tm2' => SOME (tm1 $ tm2')
| NONE => NONE)
end)
| rew2 bounds skel (Abs body) =
let
val (abs, tm') = variant_absfree bounds body;
val skel' = (case skel of Abs (_, _, skel') => skel' | _ => skel0)
in case rew1 (bounds + 1) skel' tm' of
SOME tm'' => SOME (abs tm'')
| NONE => NONE
end
| rew2 _ _ _ = NONE;
in if_none (rew1 0 skel0 tm) tm end;
end;
val trace_unify_fail = Pattern.trace_unify_fail;