(* Title: Pure/pattern.ML ID: $Id: pattern.ML,v 1.47 2005/09/15 15:16:58 wenzelm Exp $ 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 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;