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(* Title: Pure/pattern.ML
Author: Tobias Nipkow, Christine Heinzelmann, and Stefan Berghofer, 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
exception Unif
exception Pattern
val unify_trace_failure: bool Config.T
val unify_types: Context.generic -> typ * typ -> Envir.env -> Envir.env
val unify: Context.generic -> term * term -> Envir.env -> Envir.env
exception MATCH
val match: theory -> term * term -> Type.tyenv * Envir.tenv -> Type.tyenv * Envir.tenv
val first_order_match: theory -> term * term
-> Type.tyenv * Envir.tenv -> Type.tyenv * Envir.tenv
val pattern: term -> bool
end;
structure Pattern: PATTERN =
struct
exception Unif;
exception Pattern;
val unify_trace_failure = Config.declare_bool ("unify_trace_failure", \<^here>) (K false);
fun string_of_term ctxt env binders t =
Syntax.string_of_term ctxt (Envir.norm_term env (subst_bounds (map Free binders, t)));
fun bname binders i = fst (nth binders i);
fun bnames binders is = space_implode " " (map (bname binders) is);
fun typ_clash context (tye,T,U) =
if Config.get_generic context unify_trace_failure then
let
val ctxt = Context.proof_of context;
val t = Syntax.string_of_typ ctxt (Envir.norm_type tye T);
val u = Syntax.string_of_typ ctxt (Envir.norm_type tye U);
in tracing ("The following types do not unify:\n" ^ t ^ "\n" ^ u) end
else ();
fun clash context a b =
if Config.get_generic context unify_trace_failure
then tracing ("Clash: " ^ a ^ " =/= " ^ b) else ();
fun boundVar binders i =
"bound variable " ^ bname binders i ^ " (depth " ^ string_of_int i ^ ")";
fun clashBB context binders i j =
if Config.get_generic context unify_trace_failure
then clash context (boundVar binders i) (boundVar binders j) else ();
fun clashB context binders i s =
if Config.get_generic context unify_trace_failure
then clash context (boundVar binders i) s else ();
fun proj_fail context (env,binders,F,_,is,t) =
if Config.get_generic context unify_trace_failure then
let
val ctxt = Context.proof_of context
val f = Term.string_of_vname F
val xs = bnames binders is
val u = string_of_term ctxt env binders t
val ys = bnames binders (subtract (op =) is (loose_bnos t))
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 context (F,t,binders,env) =
if Config.get_generic context unify_trace_failure then
let
val ctxt = Context.proof_of context
val f = Term.string_of_vname F
val u = string_of_term ctxt 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)$(mpb d u2)
| mpb _ atom = atom
in mpb 0 end;
fun idx [] j = raise Unif
| idx(i::is) j = if (i:int) =j then length is else idx is j;
fun mkabs (binders,is,t) =
let fun mk(i::is) = let val (x,T) = nth binders i
in Abs(x,T,mk is) end
| mk [] = t
in mk is end;
val incr = mapbnd (fn i => i+1);
fun ints_of [] = []
| ints_of (Bound i ::bs) =
let val is = ints_of bs
in if member (op =) is i then raise Pattern else i::is end
| ints_of _ = raise Pattern;
fun ints_of' env ts = ints_of (map (Envir.head_norm env) ts);
fun app (s,(i::is)) = app (s$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 (nth 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 _ = raise Fail "split_type";
fun type_of_G env (T, n, is) =
let
val tyenv = Envir.type_env env;
val (Ts, U) = split_type (Envir.norm_type tyenv T, n, []);
in map (nth Ts) is ---> U end;
fun mk_hnf (binders,is,G,js) = mkabs (binders, is, app(G,js));
fun mk_new_hnf(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), mk_hnf (binders, is, G, js)) env' end;
(*predicate: downto0 (is, n) <=> is = [n, n - 1, ..., 0]*)
fun downto0 (i :: is, n) = i = n andalso downto0 (is, n - 1)
| downto0 ([], n) = n = ~1;
(*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 is_some 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 = map_filter 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), mk_hnf (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 _ = raise Fail "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 mk_new_hnf(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 subset (op =) (js, 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 = inter (op =) js is
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 Term_Ord.indexname_ord (G,F) = LESS then ff(F,Fty,is,G,Gty,js) else ff(G,Gty,js,F,Fty,is) end
fun unify_types context (T, U) (env as Envir.Envir {maxidx, tenv, tyenv}) =
if T = U then env
else
let
val thy = Context.theory_of context
val (tyenv', maxidx') = Sign.typ_unify thy (U, T) (tyenv, maxidx)
in Envir.Envir {maxidx = maxidx', tenv = tenv, tyenv = tyenv'} end
handle Type.TUNIFY => (typ_clash context (tyenv, T, U); raise Unif);
fun unif context binders (s,t) env = 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 context ((name,Ts)::binders) (ts,tt) (unify_types context (Ts, Tt) env) end
| (Abs(ns,Ts,ts),t) => unif context ((ns,Ts)::binders) (ts,(incr t)$Bound(0)) env
| (t,Abs(nt,Tt,tt)) => unif context ((nt,Tt)::binders) ((incr t)$Bound(0),tt) env
| p => cases context (binders,env,p)
and cases context (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 context (env,binders,F,Fty,ints_of' env ss,t)
| (_,(Var(F,Fty),ts)) => flexrigid context (env,binders,F,Fty,ints_of' env ts,s)
| ((Const c,ss),(Const d,ts)) => rigidrigid context (env,binders,c,d,ss,ts)
| ((Free(f),ss),(Free(g),ts)) => rigidrigid context (env,binders,f,g,ss,ts)
| ((Bound(i),ss),(Bound(j),ts)) => rigidrigidB context (env,binders,i,j,ss,ts)
| ((Abs(_),_),_) => raise Pattern
| (_,(Abs(_),_)) => raise Pattern
| ((Const(c,_),_),(Free(f,_),_)) => (clash context c f; raise Unif)
| ((Const(c,_),_),(Bound i,_)) => (clashB context binders i c; raise Unif)
| ((Free(f,_),_),(Const(c,_),_)) => (clash context f c; raise Unif)
| ((Free(f,_),_),(Bound i,_)) => (clashB context binders i f; raise Unif)
| ((Bound i,_),(Const(c,_),_)) => (clashB context binders i c; raise Unif)
| ((Bound i,_),(Free(f,_),_)) => (clashB context binders i f; raise Unif)
and rigidrigid context (env,binders,(a,Ta),(b,Tb),ss,ts) =
if a<>b then (clash context a b; raise Unif)
else env |> unify_types context (Ta,Tb) |> fold (unif context binders) (ss~~ts)
and rigidrigidB context (env,binders,i,j,ss,ts) =
if i <> j then (clashBB context binders i j; raise Unif)
else fold (unif context binders) (ss~~ts) env
and flexrigid context (params as (env,binders,F,Fty,is,t)) =
if occurs(F,t,env) then (ocheck_fail context (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 context params; raise Unif));
fun unify context = unif context [];
(*** Matching ***)
exception MATCH;
fun typ_match thy TU tyenv = Sign.typ_match thy TU tyenv
handle Type.TYPE_MATCH => raise MATCH;
(*First-order matching;
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!
Types are matched on the fly.
The parameter inAbs is an optimization to avoid calling is_open;
it has the funny consequence that outside abstractions
?x matches terms containing loose Bounds.
*)
fun first_order_match thy =
let
fun mtch inAbs (instsp as (tyinsts,insts)) = fn
(Var(ixn,T), t) =>
if inAbs andalso Term.is_open t then raise MATCH
else (case Envir.lookup1 insts (ixn, T) of
NONE => (typ_match thy (T, fastype_of t) tyinsts,
Vartab.update_new (ixn, (T, t)) insts)
| SOME u => if Envir.aeconv (t, u) then instsp else raise MATCH)
| (Free (a,T), Free (b,U)) =>
if a=b then (typ_match thy (T,U) tyinsts, insts) else raise MATCH
| (Const (a,T), Const (b,U)) =>
if a=b then (typ_match thy (T,U) tyinsts, insts) else raise MATCH
| (Bound i, Bound j) => if i=j then instsp else raise MATCH
| (Abs(_,T,t), Abs(_,U,u)) =>
mtch true (typ_match thy (T,U) tyinsts, insts) (t,u)
| (f$t, g$u) => mtch inAbs (mtch inAbs instsp (f,g)) (t, u)
| (t, Abs(_,U,u)) => mtch true instsp ((incr t)$(Bound 0), u)
| _ => raise MATCH
in fn tu => fn env => mtch false env tu end;
(* Matching of higher-order patterns *)
fun match_bind(itms,binders,ixn,T,is,t) =
let val js = loose_bnos t
in if null is
then if null js then Vartab.update_new (ixn, (T, t)) itms else raise MATCH
else if subset (op =) (js, is)
then let val t' = if downto0(is,length binders - 1) then t
else mapbnd (idx is) t
in Vartab.update_new (ixn, (T, mkabs (binders, is, t'))) itms end
else raise MATCH
end;
fun match thy (po as (pat,obj)) envir =
let
(* Pre: pat and obj have same type *)
fun mtch binders (pat,obj) (env as (iTs,itms)) =
case pat of
Abs(ns,Ts,ts) =>
(case obj of
Abs(nt,Tt,tt) => mtch ((nt,Tt)::binders) (ts,tt) env
| _ => let val Tt = Envir.subst_type iTs Ts
in mtch((ns,Tt)::binders) (ts,(incr obj)$Bound(0)) env end)
| _ => (case obj of
Abs(nt,Tt,tt) =>
mtch((nt,Tt)::binders) ((incr pat)$Bound(0),tt) env
| _ => 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) = fold (mtch binders) (pargs~~oargs) (iTs,itms)
handle ListPair.UnequalLengths => raise MATCH
fun rigrig2((a:string,Ta),(b,Tb),oargs) =
if a <> b then raise MATCH
else rigrig1(typ_match thy (Ta,Tb) iTs, oargs)
in case ph of
Var(ixn,T) =>
let val is = ints_of pargs
in case Envir.lookup1 itms (ixn, T) of
NONE => (iTs,match_bind(itms,binders,ixn,T,is,obj))
| SOME u => if Envir.aeconv (obj, 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 envir' = apfst (typ_match thy (pT, oT)) envir;
in mtch [] po envir' handle Pattern => first_order_match thy po envir' end;
fun pattern (Abs (_, _, t)) = pattern t
| pattern t =
let val (head, args) = strip_comb t in
if is_Var head then
forall is_Bound args andalso not (has_duplicates (op aconv) args)
else forall pattern args
end;
end;
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