src/Pure/unify.ML

Comment

(*  Title:      Pure/unify.ML
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   Cambridge University 1992

Higher-Order Unification.

Types as well as terms are unified.  The outermost functions assume
the terms to be unified already have the same type.  In resolution,
this is assured because both have type "prop".
*)

signature UNIFY =
sig
  val trace_bound: int ref
  val trace_simp: bool ref
  val trace_types: bool ref
  val search_bound: int ref
  val unifiers: theory * Envir.env * ((term * term) list) ->
    (Envir.env * (term * term) list) Seq.seq
  val smash_unifiers: theory -> (term * term) list -> Envir.env -> Envir.env Seq.seq
  val matchers: theory -> (term * term) list -> Envir.env Seq.seq
  val matches_list: theory -> term list -> term list -> bool
end

structure Unify : UNIFY =
struct

(*Unification options*)

val trace_bound = ref 25  (*tracing starts above this depth, 0 for full*)
and search_bound = ref 30 (*unification quits above this depth*)
and trace_simp = ref false  (*print dpairs before calling SIMPL*)
and trace_types = ref false (*announce potential incompleteness of type unification*)

type binderlist = (string*typ) list;

type dpair = binderlist * term * term;

fun body_type(Envir.Envir{iTs,...}) =
let fun bT(Type("fun",[_,T])) = bT T
      | bT(T as TVar ixnS) = (case Type.lookup (iTs, ixnS) of
    NONE => T | SOME(T') => bT T')
      | bT T = T
in bT end;

fun binder_types(Envir.Envir{iTs,...}) =
let fun bTs(Type("fun",[T,U])) = T :: bTs U
      | bTs(T as TVar ixnS) = (case Type.lookup (iTs, ixnS) of
    NONE => [] | SOME(T') => bTs T')
      | bTs _ = []
in bTs end;

fun strip_type env T = (binder_types env T, body_type env T);

fun fastype env (Ts, t) = Envir.fastype env (map snd Ts) t;


(*Eta normal form*)
fun eta_norm(env as Envir.Envir{iTs,...}) =
  let fun etif (Type("fun",[T,U]), t) =
      Abs("", T, etif(U, incr_boundvars 1 t $ Bound 0))
  | etif (TVar ixnS, t) =
      (case Type.lookup (iTs, ixnS) of
      NONE => t | SOME(T) => etif(T,t))
  | etif (_,t) = t;
      fun eta_nm (rbinder, Abs(a,T,body)) =
      Abs(a, T, eta_nm ((a,T)::rbinder, body))
  | eta_nm (rbinder, t) = etif(fastype env (rbinder,t), t)
  in eta_nm end;


(*OCCURS CHECK
  Does the uvar occur in the term t?
  two forms of search, for whether there is a rigid path to the current term.
  "seen" is list of variables passed thru, is a memo variable for sharing.
  This version searches for nonrigid occurrence, returns true if found.
  Since terms may contain variables with same name and different types,
  the occurs check must ignore the types of variables. This avoids
  that ?x::?'a is unified with f(?x::T), which may lead to a cyclic
  substitution when ?'a is instantiated with T later. *)
fun occurs_terms (seen: (indexname list) ref,
      env: Envir.env, v: indexname, ts: term list): bool =
  let fun occurs [] = false
  | occurs (t::ts) =  occur t  orelse  occurs ts
      and occur (Const _)  = false
  | occur (Bound _)  = false
  | occur (Free _)  = false
  | occur (Var (w, T))  =
      if member (op =) (!seen) w then false
      else if eq_ix(v,w) then true
        (*no need to lookup: v has no assignment*)
      else (seen := w:: !seen;
            case Envir.lookup (env, (w, T)) of
          NONE    => false
        | SOME t => occur t)
  | occur (Abs(_,_,body)) = occur body
  | occur (f$t) = occur t  orelse   occur f
  in  occurs ts  end;



(* f(a1,...,an)  ---->   (f,  [a1,...,an])  using the assignments*)
fun head_of_in (env,t) : term = case t of
    f$_ => head_of_in(env,f)
  | Var vT => (case Envir.lookup (env, vT) of
      SOME u => head_of_in(env,u)  |  NONE   => t)
  | _ => t;


datatype occ = NoOcc | Nonrigid | Rigid;

(* Rigid occur check
Returns Rigid    if it finds a rigid occurrence of the variable,
        Nonrigid if it finds a nonrigid path to the variable.
        NoOcc    otherwise.
  Continues searching for a rigid occurrence even if it finds a nonrigid one.

Condition for detecting non-unifable terms: [ section 5.3 of Huet (1975) ]
   a rigid path to the variable, appearing with no arguments.
Here completeness is sacrificed in order to reduce danger of divergence:
   reject ALL rigid paths to the variable.
Could check for rigid paths to bound variables that are out of scope.
Not necessary because the assignment test looks at variable's ENTIRE rbinder.

Treatment of head(arg1,...,argn):
If head is a variable then no rigid path, switch to nonrigid search
for arg1,...,argn.
If head is an abstraction then possibly no rigid path (head could be a
   constant function) so again use nonrigid search.  Happens only if
   term is not in normal form.

Warning: finds a rigid occurrence of ?f in ?f(t).
  Should NOT be called in this case: there is a flex-flex unifier
*)
fun rigid_occurs_term (seen: (indexname list)ref, env, v: indexname, t) =
  let fun nonrigid t = if occurs_terms(seen,env,v,[t]) then Nonrigid
           else NoOcc
      fun occurs [] = NoOcc
  | occurs (t::ts) =
            (case occur t of
               Rigid => Rigid
             | oc =>  (case occurs ts of NoOcc => oc  |  oc2 => oc2))
      and occomb (f$t) =
            (case occur t of
               Rigid => Rigid
             | oc =>  (case occomb f of NoOcc => oc  |  oc2 => oc2))
        | occomb t = occur t
      and occur (Const _)  = NoOcc
  | occur (Bound _)  = NoOcc
  | occur (Free _)  = NoOcc
  | occur (Var (w, T))  =
      if member (op =) (!seen) w then NoOcc
      else if eq_ix(v,w) then Rigid
      else (seen := w:: !seen;
            case Envir.lookup (env, (w, T)) of
          NONE    => NoOcc
        | SOME t => occur t)
  | occur (Abs(_,_,body)) = occur body
  | occur (t as f$_) =  (*switch to nonrigid search?*)
     (case head_of_in (env,f) of
        Var (w,_) => (*w is not assigned*)
    if eq_ix(v,w) then Rigid
    else  nonrigid t
      | Abs(_,_,body) => nonrigid t (*not in normal form*)
      | _ => occomb t)
  in  occur t  end;


exception CANTUNIFY;  (*Signals non-unifiability.  Does not signal errors!*)
exception ASSIGN; (*Raised if not an assignment*)


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 => raise CANTUNIFY;

fun test_unify_types thy (args as (T,U,_)) =
let val str_of = Sign.string_of_typ thy;
    fun warn() = tracing ("Potential loss of completeness: " ^ str_of U ^ " = " ^ str_of T);
    val env' = unify_types thy args
in if is_TVar(T) orelse is_TVar(U) then warn() else ();
   env'
end;

(*Is the term eta-convertible to a single variable with the given rbinder?
  Examples: ?a   ?f(B.0)   ?g(B.1,B.0)
  Result is var a for use in SIMPL. *)
fun get_eta_var ([], _, Var vT)  =  vT
  | get_eta_var (_::rbinder, n, f $ Bound i) =
  if  n=i  then  get_eta_var (rbinder, n+1, f)
     else  raise ASSIGN
  | get_eta_var _ = raise ASSIGN;


(*Solve v=u by assignment -- "fixedpoint" to Huet -- if v not in u.
  If v occurs rigidly then nonunifiable.
  If v occurs nonrigidly then must use full algorithm. *)
fun assignment thy (env, rbinder, t, u) =
    let val vT as (v,T) = get_eta_var (rbinder, 0, t)
    in  case rigid_occurs_term (ref [], env, v, u) of
        NoOcc => let val env = unify_types thy (body_type env T,
             fastype env (rbinder,u),env)
    in Envir.update ((vT, Logic.rlist_abs (rbinder, u)), env) end
      | Nonrigid =>  raise ASSIGN
      | Rigid =>  raise CANTUNIFY
    end;


(*Extends an rbinder with a new disagreement pair, if both are abstractions.
  Tries to unify types of the bound variables!
  Checks that binders have same length, since terms should be eta-normal;
    if not, raises TERM, probably indicating type mismatch.
  Uses variable a (unless the null string) to preserve user's naming.*)
fun new_dpair thy (rbinder, Abs(a,T,body1), Abs(b,U,body2), env) =
  let val env' = unify_types thy (T,U,env)
      val c = if a="" then b else a
  in new_dpair thy ((c,T) :: rbinder, body1, body2, env') end
    | new_dpair _ (_, Abs _, _, _) = raise TERM ("new_dpair", [])
    | new_dpair _ (_, _, Abs _, _) = raise TERM ("new_dpair", [])
    | new_dpair _ (rbinder, t1, t2, env) = ((rbinder, t1, t2), env);


fun head_norm_dpair thy (env, (rbinder,t,u)) : dpair * Envir.env =
     new_dpair thy (rbinder,
    eta_norm env (rbinder, Envir.head_norm env t),
      eta_norm env (rbinder, Envir.head_norm env u), env);



(*flexflex: the flex-flex pairs,  flexrigid: the flex-rigid pairs
  Does not perform assignments for flex-flex pairs:
    may create nonrigid paths, which prevent other assignments.
  Does not even identify Vars in dpairs such as ?a =?= ?b; an attempt to
    do so caused numerous problems with no compensating advantage.
*)
fun SIMPL0 thy (dp0, (env,flexflex,flexrigid))
  : Envir.env * dpair list * dpair list =
    let val (dp as (rbinder,t,u), env) = head_norm_dpair thy (env,dp0);
      fun SIMRANDS(f$t, g$u, env) =
      SIMPL0 thy ((rbinder,t,u), SIMRANDS(f,g,env))
        | SIMRANDS (t as _$_, _, _) =
    raise TERM ("SIMPL: operands mismatch", [t,u])
        | SIMRANDS (t, u as _$_, _) =
    raise TERM ("SIMPL: operands mismatch", [t,u])
        | SIMRANDS(_,_,env) = (env,flexflex,flexrigid);
    in case (head_of t, head_of u) of
       (Var(_,T), Var(_,U)) =>
      let val T' = body_type env T and U' = body_type env U;
    val env = unify_types thy (T',U',env)
      in (env, dp::flexflex, flexrigid) end
     | (Var _, _) =>
      ((assignment thy (env,rbinder,t,u), flexflex, flexrigid)
       handle ASSIGN => (env, flexflex, dp::flexrigid))
     | (_, Var _) =>
      ((assignment thy (env,rbinder,u,t), flexflex, flexrigid)
       handle ASSIGN => (env, flexflex, (rbinder,u,t)::flexrigid))
     | (Const(a,T), Const(b,U)) =>
      if a=b then SIMRANDS(t,u, unify_types thy (T,U,env))
      else raise CANTUNIFY
     | (Bound i,    Bound j)    =>
      if i=j  then SIMRANDS(t,u,env) else raise CANTUNIFY
     | (Free(a,T),  Free(b,U))  =>
      if a=b then SIMRANDS(t,u, unify_types thy (T,U,env))
      else raise CANTUNIFY
     | _ => raise CANTUNIFY
    end;


(* changed(env,t) checks whether the head of t is a variable assigned in env*)
fun changed (env, f$_) = changed (env,f)
  | changed (env, Var v) =
      (case Envir.lookup(env,v) of NONE=>false  |  _ => true)
  | changed _ = false;


(*Recursion needed if any of the 'head variables' have been updated
  Clever would be to re-do just the affected dpairs*)
fun SIMPL thy (env,dpairs) : Envir.env * dpair list * dpair list =
    let val all as (env',flexflex,flexrigid) =
      foldr (SIMPL0 thy) (env,[],[]) dpairs;
  val dps = flexrigid@flexflex
    in if exists (fn ((_,t,u)) => changed(env',t) orelse changed(env',u)) dps
       then SIMPL thy (env',dps) else all
    end;


(*Makes the terms E1,...,Em,    where Ts = [T...Tm].
  Each Ei is   ?Gi(B.(n-1),...,B.0), and has type Ti
  The B.j are bound vars of binder.
  The terms are not made in eta-normal-form, SIMPL does that later.
  If done here, eta-expansion must be recursive in the arguments! *)
fun make_args name (binder: typ list, env, []) = (env, [])   (*frequent case*)
  | make_args name (binder: typ list, env, Ts) : Envir.env * term list =
       let fun funtype T = binder--->T;
     val (env', vars) = Envir.genvars name (env, map funtype Ts)
       in  (env',  map (fn var=> Logic.combound(var, 0, length binder)) vars)  end;


(*Abstraction over a list of types, like list_abs*)
fun types_abs ([],u) = u
  | types_abs (T::Ts, u) = Abs("", T, types_abs(Ts,u));

(*Abstraction over the binder of a type*)
fun type_abs (env,T,t) = types_abs(binder_types env T, t);


(*MATCH taking "big steps".
  Copies u into the Var v, using projection on targs or imitation.
  A projection is allowed unless SIMPL raises an exception.
  Allocates new variables in projection on a higher-order argument,
    or if u is a variable (flex-flex dpair).
  Returns long sequence of every way of copying u, for backtracking
  For example, projection in ?b'(?a) may be wrong if other dpairs constrain ?a.
  The order for trying projections is crucial in ?b'(?a)
  NB "vname" is only used in the call to make_args!!   *)
fun matchcopy thy vname = let fun mc(rbinder, targs, u, ed as (env,dpairs))
  : (term * (Envir.env * dpair list))Seq.seq =
let (*Produce copies of uarg and cons them in front of uargs*)
    fun copycons uarg (uargs, (env, dpairs)) =
  Seq.map(fn (uarg', ed') => (uarg'::uargs, ed'))
      (mc (rbinder, targs,eta_norm env (rbinder, Envir.head_norm env uarg),
     (env, dpairs)));
  (*Produce sequence of all possible ways of copying the arg list*)
    fun copyargs [] = Seq.cons ([],ed) Seq.empty
      | copyargs (uarg::uargs) = Seq.maps (copycons uarg) (copyargs uargs);
    val (uhead,uargs) = strip_comb u;
    val base = body_type env (fastype env (rbinder,uhead));
    fun joinargs (uargs',ed') = (list_comb(uhead,uargs'), ed');
    (*attempt projection on argument with given typ*)
    val Ts = map (curry (fastype env) rbinder) targs;
    fun projenv (head, (Us,bary), targ, tail) =
  let val env = if !trace_types then test_unify_types thy (base,bary,env)
          else unify_types thy (base,bary,env)
  in Seq.make (fn () =>
      let val (env',args) = make_args vname (Ts,env,Us);
    (*higher-order projection: plug in targs for bound vars*)
    fun plugin arg = list_comb(head_of arg, targs);
    val dp = (rbinder, list_comb(targ, map plugin args), u);
    val (env2,frigid,fflex) = SIMPL thy (env', dp::dpairs)
        (*may raise exception CANTUNIFY*)
      in  SOME ((list_comb(head,args), (env2, frigid@fflex)),
      tail)
      end  handle CANTUNIFY => Seq.pull tail)
  end handle CANTUNIFY => tail;
    (*make a list of projections*)
    fun make_projs (T::Ts, targ::targs) =
        (Bound(length Ts), T, targ) :: make_projs (Ts,targs)
      | make_projs ([],[]) = []
      | make_projs _ = raise TERM ("make_projs", u::targs);
    (*try projections and imitation*)
    fun matchfun ((bvar,T,targ)::projs) =
         (projenv(bvar, strip_type env T, targ, matchfun projs))
      | matchfun [] = (*imitation last of all*)
        (case uhead of
     Const _ => Seq.map joinargs (copyargs uargs)
         | Free _  => Seq.map joinargs (copyargs uargs)
         | _ => Seq.empty)  (*if Var, would be a loop!*)
in case uhead of
  Abs(a, T, body) =>
      Seq.map(fn (body', ed') => (Abs (a,T,body'), ed'))
    (mc ((a,T)::rbinder,
      (map (incr_boundvars 1) targs) @ [Bound 0], body, ed))
      | Var (w,uary) =>
      (*a flex-flex dpair: make variable for t*)
      let val (env', newhd) = Envir.genvar (#1 w) (env, Ts---> base)
    val tabs = Logic.combound(newhd, 0, length Ts)
    val tsub = list_comb(newhd,targs)
      in  Seq.single (tabs, (env', (rbinder,tsub,u):: dpairs))
      end
      | _ =>  matchfun(rev(make_projs(Ts, targs)))
end
in mc end;


(*Call matchcopy to produce assignments to the variable in the dpair*)
fun MATCH thy (env, (rbinder,t,u), dpairs)
  : (Envir.env * dpair list)Seq.seq =
  let val (Var (vT as (v, T)), targs) = strip_comb t;
      val Ts = binder_types env T;
      fun new_dset (u', (env',dpairs')) =
    (*if v was updated to s, must unify s with u' *)
    case Envir.lookup (env', vT) of
        NONE => (Envir.update ((vT, types_abs(Ts, u')), env'),  dpairs')
      | SOME s => (env', ([], s, types_abs(Ts, u'))::dpairs')
  in Seq.map new_dset
         (matchcopy thy (#1 v) (rbinder, targs, u, (env,dpairs)))
  end;



(**** Flex-flex processing ****)

(*At end of unification, do flex-flex assignments like ?a -> ?f(?b)
  Attempts to update t with u, raising ASSIGN if impossible*)
fun ff_assign thy (env, rbinder, t, u) : Envir.env =
let val vT as (v,T) = get_eta_var(rbinder,0,t)
in if occurs_terms (ref [], env, v, [u]) then raise ASSIGN
   else let val env = unify_types thy (body_type env T,
          fastype env (rbinder,u),
          env)
  in Envir.vupdate ((vT, Logic.rlist_abs (rbinder, u)), env) end
end;


(*Flex argument: a term, its type, and the index that refers to it.*)
type flarg = {t: term,  T: typ,  j: int};


(*Form the arguments into records for deletion/sorting.*)
fun flexargs ([],[],[]) = [] : flarg list
  | flexargs (j::js, t::ts, T::Ts) = {j=j, t=t, T=T} :: flexargs(js,ts,Ts)
  | flexargs _ = error"flexargs";


(*If an argument contains a banned Bound, then it should be deleted.
  But if the only path is flexible, this is difficult; the code gives up!
  In  %x y.?a(x) =?= %x y.?b(?c(y)) should we instantiate ?b or ?c *)
exception CHANGE_FAIL;   (*flexible occurrence of banned variable*)


(*Check whether the 'banned' bound var indices occur rigidly in t*)
fun rigid_bound (lev, banned) t =
  let val (head,args) = strip_comb t
  in
      case head of
    Bound i => member (op =) banned (i-lev)  orelse
               exists (rigid_bound (lev, banned)) args
  | Var _ => false  (*no rigid occurrences here!*)
  | Abs (_,_,u) =>
         rigid_bound(lev+1, banned) u  orelse
         exists (rigid_bound (lev, banned)) args
  | _ => exists (rigid_bound (lev, banned)) args
  end;

(*Squash down indices at level >=lev to delete the banned from a term.*)
fun change_bnos banned =
  let fun change lev (Bound i) =
      if i<lev then Bound i
      else  if member (op =) banned (i-lev)
      then raise CHANGE_FAIL (**flexible occurrence: give up**)
      else  Bound (i - length (List.filter (fn j => j < i-lev) banned))
  | change lev (Abs (a,T,t)) = Abs (a, T, change(lev+1) t)
  | change lev (t$u) = change lev t $ change lev u
  | change lev t = t
  in  change 0  end;

(*Change indices, delete the argument if it contains a banned Bound*)
fun change_arg banned ({j,t,T}, args) : flarg list =
    if rigid_bound (0, banned) t  then  args  (*delete argument!*)
    else  {j=j, t= change_bnos banned t, T=T} :: args;


(*Sort the arguments to create assignments if possible:
  create eta-terms like ?g(B.1,B.0) *)
fun arg_less ({t= Bound i1,...}, {t= Bound i2,...}) = (i2<i1)
  | arg_less (_:flarg, _:flarg) = false;

(*Test whether the new term would be eta-equivalent to a variable --
  if so then there is no point in creating a new variable*)
fun decreasing n ([]: flarg list) = (n=0)
  | decreasing n ({j,...}::args) = j=n-1 andalso decreasing (n-1) args;

(*Delete banned indices in the term, simplifying it.
  Force an assignment, if possible, by sorting the arguments.
  Update its head; squash indices in arguments. *)
fun clean_term banned (env,t) =
    let val (Var(v,T), ts) = strip_comb t
  val (Ts,U) = strip_type env T
  and js = length ts - 1  downto 0
  val args = sort (make_ord arg_less)
    (foldr (change_arg banned) [] (flexargs (js,ts,Ts)))
  val ts' = map (#t) args
    in
    if decreasing (length Ts) args then (env, (list_comb(Var(v,T), ts')))
    else let val (env',v') = Envir.genvar (#1v) (env, map (#T) args ---> U)
       val body = list_comb(v', map (Bound o #j) args)
       val env2 = Envir.vupdate ((((v, T), types_abs(Ts, body)),   env'))
       (*the vupdate affects ts' if they contain v*)
   in
       (env2, Envir.norm_term env2 (list_comb(v',ts')))
         end
    end;


(*Add tpair if not trivial or already there.
  Should check for swapped pairs??*)
fun add_tpair (rbinder, (t0,u0), tpairs) : (term*term) list =
  if t0 aconv u0 then tpairs
  else
  let val t = Logic.rlist_abs(rbinder, t0)  and  u = Logic.rlist_abs(rbinder, u0);
      fun same(t',u') = (t aconv t') andalso (u aconv u')
  in  if exists same tpairs  then tpairs  else (t,u)::tpairs  end;


(*Simplify both terms and check for assignments.
  Bound vars in the binder are "banned" unless used in both t AND u *)
fun clean_ffpair thy ((rbinder, t, u), (env,tpairs)) =
  let val loot = loose_bnos t  and  loou = loose_bnos u
      fun add_index (((a,T), j), (bnos, newbinder)) =
            if  member (op =) loot j  andalso  member (op =) loou j
            then  (bnos, (a,T)::newbinder)  (*needed by both: keep*)
            else  (j::bnos, newbinder);   (*remove*)
      val indices = 0 upto (length rbinder - 1);
      val (banned,rbin') = foldr add_index ([],[]) (rbinder~~indices);
      val (env', t') = clean_term banned (env, t);
      val (env'',u') = clean_term banned (env',u)
  in  (ff_assign thy (env'', rbin', t', u'), tpairs)
      handle ASSIGN => (ff_assign thy (env'', rbin', u', t'), tpairs)
      handle ASSIGN => (env'', add_tpair(rbin', (t',u'), tpairs))
  end
  handle CHANGE_FAIL => (env, add_tpair(rbinder, (t,u), tpairs));


(*IF the flex-flex dpair is an assignment THEN do it  ELSE  put in tpairs
  eliminates trivial tpairs like t=t, as well as repeated ones
  trivial tpairs can easily escape SIMPL:  ?A=t, ?A=?B, ?B=t gives t=t
  Resulting tpairs MAY NOT be in normal form:  assignments may occur here.*)
fun add_ffpair thy ((rbinder,t0,u0), (env,tpairs))
      : Envir.env * (term*term)list =
  let val t = Envir.norm_term env t0  and  u = Envir.norm_term env u0
  in  case  (head_of t, head_of u) of
      (Var(v,T), Var(w,U)) =>  (*Check for identical variables...*)
  if eq_ix(v,w) then     (*...occur check would falsely return true!*)
      if T=U then (env, add_tpair (rbinder, (t,u), tpairs))
      else raise TERM ("add_ffpair: Var name confusion", [t,u])
  else if Term.indexname_ord (v, w) = LESS then (*prefer to update the LARGER variable*)
       clean_ffpair thy ((rbinder, u, t), (env,tpairs))
        else clean_ffpair thy ((rbinder, t, u), (env,tpairs))
    | _ => raise TERM ("add_ffpair: Vars expected", [t,u])
  end;


(*Print a tracing message + list of dpairs.
  In t==u print u first because it may be rigid or flexible --
    t is always flexible.*)
fun print_dpairs thy msg (env,dpairs) =
  let fun pdp (rbinder,t,u) =
        let fun termT t = Sign.pretty_term thy
                              (Envir.norm_term env (Logic.rlist_abs(rbinder,t)))
            val bsymbs = [termT u, Pretty.str" =?=", Pretty.brk 1,
                          termT t];
        in tracing(Pretty.string_of(Pretty.blk(0,bsymbs))) end;
  in  tracing msg;  List.app pdp dpairs  end;


(*Unify the dpairs in the environment.
  Returns flex-flex disagreement pairs NOT IN normal form.
  SIMPL may raise exception CANTUNIFY. *)
fun hounifiers (thy,env, tus : (term*term)list)
  : (Envir.env * (term*term)list)Seq.seq =
  let fun add_unify tdepth ((env,dpairs), reseq) =
    Seq.make (fn()=>
    let val (env',flexflex,flexrigid) =
         (if tdepth> !trace_bound andalso !trace_simp
    then print_dpairs thy "Enter SIMPL" (env,dpairs)  else ();
    SIMPL thy (env,dpairs))
    in case flexrigid of
        [] => SOME (foldr (add_ffpair thy) (env',[]) flexflex, reseq)
      | dp::frigid' =>
    if tdepth > !search_bound then
        (warning "Unification bound exceeded"; Seq.pull reseq)
    else
    (if tdepth > !trace_bound then
        print_dpairs thy "Enter MATCH" (env',flexrigid@flexflex)
     else ();
     Seq.pull (Seq.it_right (add_unify (tdepth+1))
         (MATCH thy (env',dp, frigid'@flexflex), reseq)))
    end
    handle CANTUNIFY =>
      (if tdepth > !trace_bound then tracing"Failure node" else ();
       Seq.pull reseq));
     val dps = map (fn(t,u)=> ([],t,u)) tus
  in add_unify 1 ((env, dps), Seq.empty) end;

fun unifiers (params as (thy, env, tus)) =
  Seq.cons (fold (Pattern.unify thy) tus env, []) Seq.empty
    handle Pattern.Unif => Seq.empty
         | Pattern.Pattern => hounifiers params;


(*For smash_flexflex1*)
fun var_head_of (env,t) : indexname * typ =
  case head_of (strip_abs_body (Envir.norm_term env t)) of
      Var(v,T) => (v,T)
    | _ => raise CANTUNIFY;  (*not flexible, cannot use trivial substitution*)


(*Eliminate a flex-flex pair by the trivial substitution, see Huet (1975)
  Unifies ?f(t1...rm) with ?g(u1...un) by ?f -> %x1...xm.?a, ?g -> %x1...xn.?a
  Unfortunately, unifies ?f(t,u) with ?g(t,u) by ?f, ?g -> %(x,y)?a,
  though just ?g->?f is a more general unifier.
  Unlike Huet (1975), does not smash together all variables of same type --
    requires more work yet gives a less general unifier (fewer variables).
  Handles ?f(t1...rm) with ?f(u1...um) to avoid multiple updates. *)
fun smash_flexflex1 ((t,u), env) : Envir.env =
  let val vT as (v,T) = var_head_of (env,t)
      and wU as (w,U) = var_head_of (env,u);
      val (env', var) = Envir.genvar (#1v) (env, body_type env T)
      val env'' = Envir.vupdate ((wU, type_abs (env', U, var)), env')
  in  if vT = wU then env''  (*the other update would be identical*)
      else Envir.vupdate ((vT, type_abs (env', T, var)), env'')
  end;


(*Smash all flex-flexpairs.  Should allow selection of pairs by a predicate?*)
fun smash_flexflex (env,tpairs) : Envir.env =
  foldr smash_flexflex1 env tpairs;

(*Returns unifiers with no remaining disagreement pairs*)
fun smash_unifiers thy tus env =
    Seq.map smash_flexflex (unifiers(thy,env,tus));


(*Pattern matching*)
fun first_order_matchers thy pairs (Envir.Envir {asol = tenv, iTs = tyenv, maxidx}) =
  let val (tyenv', tenv') = fold (Pattern.first_order_match thy) pairs (tyenv, tenv)
  in Seq.single (Envir.Envir {asol = tenv', iTs = tyenv', maxidx = maxidx}) end
  handle Pattern.MATCH => Seq.empty;

(*General matching -- keeps variables disjoint*)
fun matchers _ [] = Seq.single (Envir.empty ~1)
  | matchers thy pairs =
      let
        val maxidx = fold (Term.maxidx_term o #2) pairs ~1;
        val offset = maxidx + 1;
        val pairs' = map (apfst (Logic.incr_indexes ([], offset))) pairs;
        val maxidx' = fold (fn (t, u) => Term.maxidx_term t #> Term.maxidx_term u) pairs' ~1;

        val pat_tvars = fold (Term.add_tvars o #1) pairs' [];
        val pat_vars = fold (Term.add_vars o #1) pairs' [];

        val decr_indexesT =
          Term.map_atyps (fn T as TVar ((x, i), S) =>
            if i > maxidx then TVar ((x, i - offset), S) else T | T => T);
        val decr_indexes =
          Term.map_types decr_indexesT #>
          Term.map_aterms (fn t as Var ((x, i), T) =>
            if i > maxidx then Var ((x, i - offset), T) else t | t => t);

        fun norm_tvar (Envir.Envir {iTs = tyenv, ...}) ((x, i), S) =
          ((x, i - offset), (S, decr_indexesT (Envir.norm_type tyenv (TVar ((x, i), S)))));
        fun norm_var (env as Envir.Envir {iTs = tyenv, ...}) ((x, i), T) =
          let
            val T' = Envir.norm_type tyenv T;
            val t' = Envir.norm_term env (Var ((x, i), T'));
          in ((x, i - offset), (decr_indexesT T', decr_indexes t')) end;

        fun result env =
          if Envir.above env maxidx then   (* FIXME proper handling of generated vars!? *)
            SOME (Envir.Envir {maxidx = maxidx,
              iTs = Vartab.make (map (norm_tvar env) pat_tvars),
              asol = Vartab.make (map (norm_var env) pat_vars)})
          else NONE;

        val empty = Envir.empty maxidx';
      in
        Seq.append
          (Seq.map_filter result (smash_unifiers thy pairs' empty))
          (first_order_matchers thy pairs empty)
      end;

fun matches_list thy ps os =
  length ps = length os andalso is_some (Seq.pull (matchers thy (ps ~~ os)));

end;