src/Pure/pattern.ML
author berghofe
Wed May 07 10:59:46 2008 +0200 (2008-05-07)
changeset 26831 6c3eec8157d3
parent 25473 812db0f215b3
child 26939 1035c89b4c02
permissions -rw-r--r--
- Removed function eta_contract_atom, which did not quite work
- Corrected and simplified function aeconv
     1 (*  Title:      Pure/pattern.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, Christine Heinzelmann, and Stefan Berghofer
     4 
     5 Unification of Higher-Order Patterns.
     6 
     7 See also:
     8 Tobias Nipkow. Functional Unification of Higher-Order Patterns.
     9 In Proceedings of the 8th IEEE Symposium Logic in Computer Science, 1993.
    10 
    11 TODO: optimize red by special-casing it
    12 *)
    13 
    14 infix aeconv;
    15 
    16 signature PATTERN =
    17 sig
    18   val trace_unify_fail: bool ref
    19   val aeconv: term * term -> bool
    20   val eta_long: typ list -> term -> term
    21   val match: theory -> term * term -> Type.tyenv * Envir.tenv -> Type.tyenv * Envir.tenv
    22   val first_order_match: theory -> term * term
    23     -> Type.tyenv * Envir.tenv -> Type.tyenv * Envir.tenv
    24   val matches: theory -> term * term -> bool
    25   val equiv: theory -> term * term -> bool
    26   val matches_subterm: theory -> term * term -> bool
    27   val unify: theory -> term * term -> Envir.env -> Envir.env
    28   val first_order: term -> bool
    29   val pattern: term -> bool
    30   val rewrite_term: theory -> (term * term) list -> (term -> term option) list -> term -> term
    31   exception Unif
    32   exception MATCH
    33   exception Pattern
    34 end;
    35 
    36 structure Pattern: PATTERN =
    37 struct
    38 
    39 exception Unif;
    40 exception Pattern;
    41 
    42 val trace_unify_fail = ref false;
    43 
    44 fun string_of_term thy env binders t = Sign.string_of_term thy
    45   (Envir.norm_term env (subst_bounds(map Free binders,t)));
    46 
    47 fun bname binders i = fst (nth binders i);
    48 fun bnames binders is = space_implode " " (map (bname binders) is);
    49 
    50 fun typ_clash thy (tye,T,U) =
    51   if !trace_unify_fail
    52   then let val t = Sign.string_of_typ thy (Envir.norm_type tye T)
    53            and u = Sign.string_of_typ thy (Envir.norm_type tye U)
    54        in tracing("The following types do not unify:\n" ^ t ^ "\n" ^ u) end
    55   else ()
    56 
    57 fun clash a b =
    58   if !trace_unify_fail then tracing("Clash: " ^ a ^ " =/= " ^ b) else ()
    59 
    60 fun boundVar binders i =
    61   "bound variable " ^ bname binders i ^ " (depth " ^ string_of_int i ^ ")";
    62 
    63 fun clashBB binders i j =
    64   if !trace_unify_fail then clash (boundVar binders i) (boundVar binders j)
    65   else ()
    66 
    67 fun clashB binders i s =
    68   if !trace_unify_fail then clash (boundVar binders i) s
    69   else ()
    70 
    71 fun proj_fail thy (env,binders,F,_,is,t) =
    72   if !trace_unify_fail
    73   then let val f = Term.string_of_vname F
    74            val xs = bnames binders is
    75            val u = string_of_term thy env binders t
    76            val ys = bnames binders (subtract (op =) is (loose_bnos t))
    77        in tracing("Cannot unify variable " ^ f ^
    78                " (depending on bound variables " ^ xs ^ ")\nwith term " ^ u ^
    79                "\nTerm contains additional bound variable(s) " ^ ys)
    80        end
    81   else ()
    82 
    83 fun ocheck_fail thy (F,t,binders,env) =
    84   if !trace_unify_fail
    85   then let val f = Term.string_of_vname F
    86            val u = string_of_term thy env binders t
    87        in tracing("Variable " ^ f ^ " occurs in term\n" ^ u ^
    88                   "\nCannot unify!\n")
    89        end
    90   else ()
    91 
    92 fun occurs(F,t,env) =
    93     let fun occ(Var (G, T))   = (case Envir.lookup (env, (G, T)) of
    94                                  SOME(t) => occ t
    95                                | NONE    => F=G)
    96           | occ(t1$t2)      = occ t1 orelse occ t2
    97           | occ(Abs(_,_,t)) = occ t
    98           | occ _           = false
    99     in occ t end;
   100 
   101 
   102 fun mapbnd f =
   103     let fun mpb d (Bound(i))     = if i < d then Bound(i) else Bound(f(i-d)+d)
   104           | mpb d (Abs(s,T,t))   = Abs(s,T,mpb(d+1) t)
   105           | mpb d ((u1 $ u2))    = (mpb d u1)$(mpb d u2)
   106           | mpb _ atom           = atom
   107     in mpb 0 end;
   108 
   109 fun idx [] j     = raise Unif
   110   | idx(i::is) j = if (i:int) =j then length is else idx is j;
   111 
   112 fun mkabs (binders,is,t)  =
   113     let fun mk(i::is) = let val (x,T) = nth binders i
   114                         in Abs(x,T,mk is) end
   115           | mk []     = t
   116     in mk is end;
   117 
   118 val incr = mapbnd (fn i => i+1);
   119 
   120 fun ints_of []             = []
   121   | ints_of (Bound i ::bs) =
   122       let val is = ints_of bs
   123       in if member (op =) is i then raise Pattern else i::is end
   124   | ints_of _              = raise Pattern;
   125 
   126 fun ints_of' env ts = ints_of (map (Envir.head_norm env) ts);
   127 
   128 
   129 fun app (s,(i::is)) = app (s$Bound(i),is)
   130   | app (s,[])      = s;
   131 
   132 fun red (Abs(_,_,s)) (i::is) js = red s is (i::js)
   133   | red t            []      [] = t
   134   | red t            is      jn = app (mapbnd (nth jn) t,is);
   135 
   136 
   137 (* split_type ([T1,....,Tn]---> T,n,[]) = ([Tn,...,T1],T) *)
   138 fun split_type (T,0,Ts)                    = (Ts,T)
   139   | split_type (Type ("fun",[T1,T2]),n,Ts) = split_type (T2,n-1,T1::Ts)
   140   | split_type _                           = error("split_type");
   141 
   142 fun type_of_G (env as Envir.Envir {iTs, ...}) (T,n,is) =
   143   let val (Ts, U) = split_type (Envir.norm_type iTs T, n, [])
   144   in map (nth Ts) is ---> U end;
   145 
   146 fun mkhnf (binders,is,G,js) = mkabs (binders, is, app(G,js));
   147 
   148 fun mknewhnf(env,binders,is,F as (a,_),T,js) =
   149   let val (env',G) = Envir.genvar a (env,type_of_G env (T,length is,js))
   150   in Envir.update (((F, T), mkhnf (binders, is, G, js)), env') end;
   151 
   152 
   153 (*predicate: downto0 (is, n) <=> is = [n, n - 1, ..., 0]*)
   154 fun downto0 (i :: is, n) = i = n andalso downto0 (is, n - 1)
   155   | downto0 ([], n) = n = ~1;
   156 
   157 (*mk_proj_list(is) = [ |is| - k | 1 <= k <= |is| and is[k] >= 0 ]*)
   158 fun mk_proj_list is =
   159     let fun mk(i::is,j) = if is_some i then j :: mk(is,j-1) else mk(is,j-1)
   160           | mk([],_)    = []
   161     in mk(is,length is - 1) end;
   162 
   163 fun proj(s,env,binders,is) =
   164     let fun trans d i = if i<d then i else (idx is (i-d))+d;
   165         fun pr(s,env,d,binders) = (case Envir.head_norm env s of
   166               Abs(a,T,t) => let val (t',env') = pr(t,env,d+1,((a,T)::binders))
   167                             in (Abs(a,T,t'),env') end
   168             | t => (case strip_comb t of
   169                 (c as Const _,ts) =>
   170                          let val (ts',env') = prs(ts,env,d,binders)
   171                          in (list_comb(c,ts'),env') end
   172                  | (f as Free _,ts) =>
   173                          let val (ts',env') = prs(ts,env,d,binders)
   174                          in (list_comb(f,ts'),env') end
   175                  | (Bound(i),ts) =>
   176                          let val j = trans d i
   177                              val (ts',env') = prs(ts,env,d,binders)
   178                          in (list_comb(Bound j,ts'),env') end
   179                  | (Var(F as (a,_),Fty),ts) =>
   180                       let val js = ints_of' env ts;
   181                           val js' = map (try (trans d)) js;
   182                           val ks = mk_proj_list js';
   183                           val ls = map_filter I js'
   184                           val Hty = type_of_G env (Fty,length js,ks)
   185                           val (env',H) = Envir.genvar a (env,Hty)
   186                           val env'' =
   187                             Envir.update (((F, Fty), mkhnf (binders, js, H, ks)), env')
   188                       in (app(H,ls),env'') end
   189                  | _  => raise Pattern))
   190         and prs(s::ss,env,d,binders) =
   191               let val (s',env1) = pr(s,env,d,binders)
   192                   val (ss',env2) = prs(ss,env1,d,binders)
   193               in (s'::ss',env2) end
   194           | prs([],env,_,_) = ([],env)
   195    in if downto0(is,length binders - 1) then (s,env)
   196       else pr(s,env,0,binders)
   197    end;
   198 
   199 
   200 (* mk_ff_list(is,js) = [ length(is) - k | 1 <= k <= |is| and is[k] = js[k] ] *)
   201 fun mk_ff_list(is,js) =
   202     let fun mk([],[],_)        = []
   203           | mk(i::is,j::js, k) = if (i:int) = j then k :: mk(is,js,k-1)
   204                                         else mk(is,js,k-1)
   205           | mk _               = error"mk_ff_list"
   206     in mk(is,js,length is-1) end;
   207 
   208 fun flexflex1(env,binders,F,Fty,is,js) =
   209   if is=js then env
   210   else let val ks = mk_ff_list(is,js)
   211        in mknewhnf(env,binders,is,F,Fty,ks) end;
   212 
   213 fun flexflex2(env,binders,F,Fty,is,G,Gty,js) =
   214   let fun ff(F,Fty,is,G as (a,_),Gty,js) =
   215             if js subset_int is
   216             then let val t= mkabs(binders,is,app(Var(G,Gty),map (idx is) js))
   217                  in Envir.update (((F, Fty), t), env) end
   218             else let val ks = is inter_int js
   219                      val Hty = type_of_G env (Fty,length is,map (idx is) ks)
   220                      val (env',H) = Envir.genvar a (env,Hty)
   221                      fun lam(is) = mkabs(binders,is,app(H,map (idx is) ks));
   222                  in Envir.update (((G, Gty), lam js), Envir.update (((F, Fty), lam is), env'))
   223                  end;
   224   in if Term.indexname_ord (G,F) = LESS then ff(F,Fty,is,G,Gty,js) else ff(G,Gty,js,F,Fty,is) end
   225 
   226 fun unify_types thy (T,U) (env as Envir.Envir{asol,iTs,maxidx}) =
   227   if T=U then env
   228   else let val (iTs',maxidx') = Sign.typ_unify thy (U, T) (iTs, maxidx)
   229        in Envir.Envir{asol=asol,maxidx=maxidx',iTs=iTs'} end
   230        handle Type.TUNIFY => (typ_clash thy (iTs,T,U); raise Unif);
   231 
   232 fun unif thy binders (s,t) env = case (Envir.head_norm env s, Envir.head_norm env t) of
   233       (Abs(ns,Ts,ts),Abs(nt,Tt,tt)) =>
   234          let val name = if ns = "" then nt else ns
   235          in unif thy ((name,Ts)::binders) (ts,tt) env end
   236     | (Abs(ns,Ts,ts),t) => unif thy ((ns,Ts)::binders) (ts,(incr t)$Bound(0)) env
   237     | (t,Abs(nt,Tt,tt)) => unif thy ((nt,Tt)::binders) ((incr t)$Bound(0),tt) env
   238     | p => cases thy (binders,env,p)
   239 
   240 and cases thy (binders,env,(s,t)) = case (strip_comb s,strip_comb t) of
   241        ((Var(F,Fty),ss),(Var(G,Gty),ts)) =>
   242          if F = G then flexflex1(env,binders,F,Fty,ints_of' env ss,ints_of' env ts)
   243                   else flexflex2(env,binders,F,Fty,ints_of' env ss,G,Gty,ints_of' env ts)
   244       | ((Var(F,Fty),ss),_)           => flexrigid thy (env,binders,F,Fty,ints_of' env ss,t)
   245       | (_,(Var(F,Fty),ts))           => flexrigid thy (env,binders,F,Fty,ints_of' env ts,s)
   246       | ((Const c,ss),(Const d,ts))   => rigidrigid thy (env,binders,c,d,ss,ts)
   247       | ((Free(f),ss),(Free(g),ts))   => rigidrigid thy (env,binders,f,g,ss,ts)
   248       | ((Bound(i),ss),(Bound(j),ts)) => rigidrigidB thy (env,binders,i,j,ss,ts)
   249       | ((Abs(_),_),_)                => raise Pattern
   250       | (_,(Abs(_),_))                => raise Pattern
   251       | ((Const(c,_),_),(Free(f,_),_)) => (clash c f; raise Unif)
   252       | ((Const(c,_),_),(Bound i,_))   => (clashB binders i c; raise Unif)
   253       | ((Free(f,_),_),(Const(c,_),_)) => (clash f c; raise Unif)
   254       | ((Free(f,_),_),(Bound i,_))    => (clashB binders i f; raise Unif)
   255       | ((Bound i,_),(Const(c,_),_))   => (clashB binders i c; raise Unif)
   256       | ((Bound i,_),(Free(f,_),_))    => (clashB binders i f; raise Unif)
   257 
   258 
   259 and rigidrigid thy (env,binders,(a,Ta),(b,Tb),ss,ts) =
   260       if a<>b then (clash a b; raise Unif)
   261       else env |> unify_types thy (Ta,Tb) |> fold (unif thy binders) (ss~~ts)
   262 
   263 and rigidrigidB thy (env,binders,i,j,ss,ts) =
   264      if i <> j then (clashBB binders i j; raise Unif)
   265      else fold (unif thy binders) (ss~~ts) env
   266 
   267 and flexrigid thy (params as (env,binders,F,Fty,is,t)) =
   268       if occurs(F,t,env) then (ocheck_fail thy (F,t,binders,env); raise Unif)
   269       else (let val (u,env') = proj(t,env,binders,is)
   270             in Envir.update (((F, Fty), mkabs (binders, is, u)), env') end
   271             handle Unif => (proj_fail thy params; raise Unif));
   272 
   273 fun unify thy = unif thy [];
   274 
   275 
   276 (* put a term into eta long beta normal form *)
   277 fun eta_long Ts (Abs (s, T, t)) = Abs (s, T, eta_long (T :: Ts) t)
   278   | eta_long Ts t = (case strip_comb t of
   279       (Abs _, _) => eta_long Ts (Envir.beta_norm t)
   280     | (u, ts) =>
   281       let
   282         val Us = binder_types (fastype_of1 (Ts, t));
   283         val i = length Us
   284       in list_abs (map (pair "x") Us,
   285         list_comb (incr_boundvars i u, map (eta_long (rev Us @ Ts))
   286           (map (incr_boundvars i) ts @ map Bound (i - 1 downto 0))))
   287       end);
   288 
   289 
   290 (*Tests whether 2 terms are alpha/eta-convertible and have same type.
   291   Note that Consts and Vars may have more than one type.*)
   292 fun t aeconv u = Envir.eta_contract t aconv Envir.eta_contract u;
   293 
   294 
   295 (*** Matching ***)
   296 
   297 exception MATCH;
   298 
   299 fun typ_match thy TU tyenv = Sign.typ_match thy TU tyenv
   300   handle Type.TYPE_MATCH => raise MATCH;
   301 
   302 (*First-order matching;
   303   The pattern and object may have variables in common.
   304   Instantiation does not affect the object, so matching ?a with ?a+1 works.
   305   Object is eta-contracted on the fly (by eta-expanding the pattern).
   306   Precondition: the pattern is already eta-contracted!
   307   Types are matched on the fly*)
   308 fun first_order_match thy =
   309   let
   310     fun mtch k (instsp as (tyinsts,insts)) = fn
   311         (Var(ixn,T), t)  =>
   312           if k > 0 andalso loose_bvar(t,0) then raise MATCH
   313           else (case Envir.lookup' (insts, (ixn, T)) of
   314                   NONE => (typ_match thy (T, fastype_of t) tyinsts,
   315                            Vartab.update_new (ixn, (T, t)) insts)
   316                 | SOME u => if t aeconv u then instsp else raise MATCH)
   317       | (Free (a,T), Free (b,U)) =>
   318           if a=b then (typ_match thy (T,U) tyinsts, insts) else raise MATCH
   319       | (Const (a,T), Const (b,U))  =>
   320           if a=b then (typ_match thy (T,U) tyinsts, insts) else raise MATCH
   321       | (Bound i, Bound j)  =>  if  i=j  then  instsp  else raise MATCH
   322       | (Abs(_,T,t), Abs(_,U,u))  =>
   323           mtch (k + 1) (typ_match thy (T,U) tyinsts, insts) (t,u)
   324       | (f$t, g$u) => mtch k (mtch k instsp (f,g)) (t, u)
   325       | (t, Abs(_,U,u))  =>  mtch (k + 1) instsp ((incr t)$(Bound 0), u)
   326       | _ => raise MATCH
   327   in fn tu => fn env => mtch 0 env tu end;
   328 
   329 
   330 (* Matching of higher-order patterns *)
   331 
   332 fun match_bind(itms,binders,ixn,T,is,t) =
   333   let val js = loose_bnos t
   334   in if null is
   335      then if null js then Vartab.update_new (ixn, (T, t)) itms else raise MATCH
   336      else if js subset_int is
   337           then let val t' = if downto0(is,length binders - 1) then t
   338                             else mapbnd (idx is) t
   339                in Vartab.update_new (ixn, (T, mkabs (binders, is, t'))) itms end
   340           else raise MATCH
   341   end;
   342 
   343 fun match thy (po as (pat,obj)) envir =
   344 let
   345   (* Pre: pat and obj have same type *)
   346   fun mtch binders (pat,obj) (env as (iTs,itms)) =
   347     case pat of
   348       Abs(ns,Ts,ts) =>
   349         (case obj of
   350            Abs(nt,Tt,tt) => mtch ((nt,Tt)::binders) (ts,tt) env
   351          | _ => let val Tt = Envir.typ_subst_TVars iTs Ts
   352                 in mtch((ns,Tt)::binders) (ts,(incr obj)$Bound(0)) env end)
   353     | _ => (case obj of
   354               Abs(nt,Tt,tt) =>
   355                 mtch((nt,Tt)::binders) ((incr pat)$Bound(0),tt) env
   356             | _ => cases(binders,env,pat,obj))
   357 
   358   and cases(binders,env as (iTs,itms),pat,obj) =
   359     let val (ph,pargs) = strip_comb pat
   360         fun rigrig1(iTs,oargs) = fold (mtch binders) (pargs~~oargs) (iTs,itms)
   361         fun rigrig2((a:string,Ta),(b,Tb),oargs) =
   362               if a <> b then raise MATCH
   363               else rigrig1(typ_match thy (Ta,Tb) iTs, oargs)
   364     in case ph of
   365          Var(ixn,T) =>
   366            let val is = ints_of pargs
   367            in case Envir.lookup' (itms, (ixn, T)) of
   368                 NONE => (iTs,match_bind(itms,binders,ixn,T,is,obj))
   369               | SOME u => if obj aeconv (red u is []) then env
   370                           else raise MATCH
   371            end
   372        | _ =>
   373            let val (oh,oargs) = strip_comb obj
   374            in case (ph,oh) of
   375                 (Const c,Const d) => rigrig2(c,d,oargs)
   376               | (Free f,Free g)   => rigrig2(f,g,oargs)
   377               | (Bound i,Bound j) => if i<>j then raise MATCH
   378                                      else rigrig1(iTs,oargs)
   379               | (Abs _, _)        => raise Pattern
   380               | (_, Abs _)        => raise Pattern
   381               | _                 => raise MATCH
   382            end
   383     end;
   384 
   385   val pT = fastype_of pat
   386   and oT = fastype_of obj
   387   val envir' = apfst (typ_match thy (pT, oT)) envir;
   388 in mtch [] po envir' handle Pattern => first_order_match thy po envir' end;
   389 
   390 fun matches thy po = (match thy po (Vartab.empty, Vartab.empty); true) handle MATCH => false;
   391 
   392 fun equiv thy (t, u) = matches thy (t, u) andalso matches thy (u, t);
   393 
   394 
   395 (* Does pat match a subterm of obj? *)
   396 fun matches_subterm thy (pat, obj) =
   397   let
   398     fun msub bounds obj = matches thy (pat, obj) orelse
   399       (case obj of
   400         Abs (x, T, t) => msub (bounds + 1) (snd (Term.dest_abs (Name.bound bounds, T, t)))
   401       | t $ u => msub bounds t orelse msub bounds u
   402       | _ => false)
   403   in msub 0 obj end;
   404 
   405 fun first_order(Abs(_,_,t)) = first_order t
   406   | first_order(t $ u) = first_order t andalso first_order u andalso
   407                          not(is_Var t)
   408   | first_order _ = true;
   409 
   410 fun pattern (Abs (_, _, t)) = pattern t
   411   | pattern t =
   412       let val (head, args) = strip_comb t in
   413         if is_Var head then
   414           forall is_Bound args andalso not (has_duplicates (op aconv) args)
   415         else forall pattern args
   416       end;
   417 
   418 
   419 (* rewriting -- simple but fast *)
   420 
   421 fun rewrite_term thy rules procs tm =
   422   let
   423     val skel0 = Bound 0;
   424 
   425     fun variant_absfree bounds (x, T, t) =
   426       let
   427         val (x', t') = Term.dest_abs (Name.bound bounds, T, t);
   428         fun abs u = Abs (x, T, abstract_over (Free (x', T), u));
   429       in (abs, t') end;
   430 
   431     fun match_rew tm (tm1, tm2) =
   432       let val rtm = the_default tm2 (Term.rename_abs tm1 tm tm2) in
   433         SOME (Envir.subst_vars (match thy (tm1, tm) (Vartab.empty, Vartab.empty)) rtm, rtm)
   434           handle MATCH => NONE
   435       end;
   436 
   437     fun rew (Abs (_, _, body) $ t) = SOME (subst_bound (t, body), skel0)
   438       | rew tm = (case get_first (match_rew tm) rules of
   439           NONE => Option.map (rpair skel0) (get_first (fn p => p tm) procs)
   440         | x => x);
   441 
   442     fun rew1 bounds (Var _) _ = NONE
   443       | rew1 bounds skel tm = (case rew2 bounds skel tm of
   444           SOME tm1 => (case rew tm1 of
   445               SOME (tm2, skel') => SOME (the_default tm2 (rew1 bounds skel' tm2))
   446             | NONE => SOME tm1)
   447         | NONE => (case rew tm of
   448               SOME (tm1, skel') => SOME (the_default tm1 (rew1 bounds skel' tm1))
   449             | NONE => NONE))
   450 
   451     and rew2 bounds skel (tm1 $ tm2) = (case tm1 of
   452             Abs (_, _, body) =>
   453               let val tm' = subst_bound (tm2, body)
   454               in SOME (the_default tm' (rew2 bounds skel0 tm')) end
   455           | _ =>
   456             let val (skel1, skel2) = (case skel of
   457                 skel1 $ skel2 => (skel1, skel2)
   458               | _ => (skel0, skel0))
   459             in case rew1 bounds skel1 tm1 of
   460                 SOME tm1' => (case rew1 bounds skel2 tm2 of
   461                     SOME tm2' => SOME (tm1' $ tm2')
   462                   | NONE => SOME (tm1' $ tm2))
   463               | NONE => (case rew1 bounds skel2 tm2 of
   464                     SOME tm2' => SOME (tm1 $ tm2')
   465                   | NONE => NONE)
   466             end)
   467       | rew2 bounds skel (Abs body) =
   468           let
   469             val (abs, tm') = variant_absfree bounds body;
   470             val skel' = (case skel of Abs (_, _, skel') => skel' | _ => skel0)
   471           in case rew1 (bounds + 1) skel' tm' of
   472               SOME tm'' => SOME (abs tm'')
   473             | NONE => NONE
   474           end
   475       | rew2 _ _ _ = NONE;
   476 
   477   in the_default tm (rew1 0 skel0 tm) end;
   478 
   479 end;
   480 
   481 val trace_unify_fail = Pattern.trace_unify_fail;