(*  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

 *)

 infix aeconv;

 signature PATTERN =

 sig

   val trace_unify_fail: bool Unsynchronized.ref

   val aeconv: term * term -> bool

   val eta_long: typ list -> term -> term

   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 matches: theory -> term * term -> bool

   val matchess: theory -> term list * term list -> bool

   val equiv: theory -> term * term -> bool

   val matches_subterm: theory -> term * term -> bool

   val unify: theory -> term * term -> Envir.env -> Envir.env

   val first_order: term -> bool

   val pattern: term -> bool

   val match_rew: theory -> term -> term * term -> (term * term) option

   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 = Unsynchronized.ref false;

 fun string_of_term thy env binders t =

   Syntax.string_of_term_global thy

     (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 thy (tye,T,U) =

   if !trace_unify_fail

   then let val t = Syntax.string_of_typ_global thy (Envir.norm_type tye T)

            and u = Syntax.string_of_typ_global 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 = Term.string_of_vname F

            val xs = bnames binders is

            val u = string_of_term thy 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 thy (F,t,binders,env) =

   if !trace_unify_fail

   then let val f = Term.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 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 _                           = error("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 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;

 (*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), 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 TermOrd.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 thy (T, U) (env as Envir.Envir {maxidx, tenv, tyenv}) =

   if T = U then env

   else

     let 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 thy (tyenv, T, U); raise Unif);

 fun unif thy 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 thy ((name,Ts)::binders) (ts,tt) env end

     | (Abs(ns,Ts,ts),t) => unif thy ((ns,Ts)::binders) (ts,(incr t)$Bound(0)) env

     | (t,Abs(nt,Tt,tt)) => unif thy ((nt,Tt)::binders) ((incr t)$Bound(0),tt) env

     | 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 env |> unify_types thy (Ta,Tb) |> fold (unif thy binders) (ss~~ts)

 and rigidrigidB thy (env,binders,i,j,ss,ts) =

      if i <> j then (clashBB binders i j; raise Unif)

      else fold (unif thy binders) (ss~~ts) env

 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 = unif thy [];

 (* 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 = t aconv u orelse

   Envir.eta_contract t aconv Envir.eta_contract u;

 (*** 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*)

 fun first_order_match thy =

   let

     fun mtch k (instsp as (tyinsts,insts)) = fn

         (Var(ixn,T), t)  =>

           if k > 0 andalso loose_bvar(t,0) then raise MATCH

           else (case Envir.lookup' (insts, (ixn, T)) of

                   NONE => (typ_match thy (T, fastype_of t) tyinsts,

                            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 (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 (k + 1) (typ_match thy (T,U) tyinsts, insts) (t,u)

       | (f$t, g$u) => mtch k (mtch k instsp (f,g)) (t, u)

       | (t, Abs(_,U,u))  =>  mtch (k + 1) instsp ((incr t)$(Bound 0), u)

       | _ => raise MATCH

   in fn tu => fn env => mtch 0 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 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)) 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)

         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.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 envir' = apfst (typ_match thy (pT, oT)) envir;

 in mtch [] po envir' handle Pattern => first_order_match thy po envir' end;

 fun matches thy po = (match thy po (Vartab.empty, Vartab.empty); true) handle MATCH => false;

 fun matchess thy pos = (fold (match thy) (op ~~ pos) (Vartab.empty, Vartab.empty); true) handle MATCH => false;

 fun equiv thy (t, u) = matches thy (t, u) andalso matches thy (u, t);

 (* 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) => msub (bounds + 1) (snd (Term.dest_abs (Name.bound bounds, T, t)))

       | t $ u => msub bounds t orelse msub bounds u

       | _ => false)

   in msub 0 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

           forall is_Bound args andalso not (has_duplicates (op aconv) args)

         else forall pattern args

       end;

 (* rewriting -- simple but fast *)

 fun match_rew thy tm (tm1, tm2) =

   let val rtm = the_default tm2 (Term.rename_abs tm1 tm tm2) in

     SOME (Envir.subst_term (match thy (tm1, tm) (Vartab.empty, Vartab.empty)) rtm, rtm)

       handle MATCH => NONE

   end;

 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 (Name.bound bounds, T, t);

         fun abs u = Abs (x, T, abstract_over (Free (x', T), u));

       in (abs, t') end;

     fun rew (Abs (_, _, body) $ t) = SOME (subst_bound (t, body), skel0)

       | rew tm =

           (case get_first (match_rew thy 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 (the_default tm2 (rew1 bounds skel' tm2))

             | NONE => SOME tm1)

         | NONE => (case rew tm of

               SOME (tm1, skel') => SOME (the_default tm1 (rew1 bounds skel' tm1))

             | NONE => NONE))

     and rew2 bounds skel (tm1 $ tm2) = (case tm1 of

             Abs (_, _, body) =>

               let val tm' = subst_bound (tm2, body)

               in SOME (the_default tm' (rew2 bounds skel0 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 the_default tm (rew1 0 skel0 tm) end;

 end;

 val trace_unify_fail = Pattern.trace_unify_fail;