(* 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 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;