path: root/pretyping/unification.ml

(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

open Pp
open Util
open Names
open Nameops
open Term
open Termops
open Namegen
open Sign
open Environ
open Evd
open Reduction
open Reductionops
open Glob_term
open Pattern
open Evarutil
open Pretype_errors
open Retyping
open Coercion.Default
open Recordops

let occur_meta_or_undefined_evar evd c =
  let rec occrec c = match kind_of_term c with
    | Meta _ -> raise Occur
    | Evar (ev,args) ->
        (match evar_body (Evd.find evd ev) with
        | Evar_defined c ->
            occrec c; Array.iter occrec args
        | Evar_empty -> raise Occur)
    | Sort s when is_sort_variable evd s -> raise Occur
    | _ -> iter_constr occrec c
  in try occrec c; false with Occur | Not_found -> true

(* if lname_typ is [xn,An;..;x1,A1] and l is a list of terms,
   gives [x1:A1]..[xn:An]c' such that c converts to ([x1:A1]..[xn:An]c' l) *)

let abstract_scheme env c l lname_typ =
  List.fold_left2
    (fun t (locc,a) (na,_,ta) ->
       let na = match kind_of_term a with Var id -> Name id | _ -> na in
(* [occur_meta ta] test removed for support of eelim/ecase but consequences
   are unclear...
       if occur_meta ta then error "cannot find a type for the generalisation"
       else *) if occur_meta a then mkLambda_name env (na,ta,t)
       else mkLambda_name env (na,ta,subst_term_occ locc a t))
    c
    (List.rev l)
    lname_typ

let abstract_list_all env evd typ c l =
  let ctxt,_ = splay_prod_n env evd (List.length l) typ in
  let l_with_all_occs = List.map (function a -> (all_occurrences,a)) l in
  let p = abstract_scheme env c l_with_all_occs ctxt in
  try
    if is_conv_leq env evd (Typing.type_of env evd p) typ then p
    else error "abstract_list_all"
  with UserError _ | Type_errors.TypeError _ ->
    error_cannot_find_well_typed_abstraction env evd p l

(**)

(* A refinement of [conv_pb]: the integers tells how many arguments
   were applied in the context of the conversion problem; if the number
   is non zero, steps of eta-expansion will be allowed
*)

type conv_pb_up_to_eta = Cumul | ConvUnderApp of int * int

let topconv = ConvUnderApp (0,0)
let of_conv_pb = function CONV -> topconv | CUMUL -> Cumul
let conv_pb_of = function ConvUnderApp _ -> CONV | Cumul -> CUMUL
let prod_pb = function ConvUnderApp _ -> topconv | pb -> pb

let opp_status = function
  | IsSuperType -> IsSubType
  | IsSubType -> IsSuperType
  | ConvUpToEta _ | UserGiven as x -> x

let add_type_status (x,y) = ((x,TypeNotProcessed),(y,TypeNotProcessed))

let extract_instance_status = function
  | Cumul -> add_type_status (IsSubType, IsSuperType)
  | ConvUnderApp (n1,n2) -> add_type_status (ConvUpToEta n1, ConvUpToEta n2)

let rec assoc_pair x = function
    [] -> raise Not_found
  | (a,b,_)::l -> if compare a x = 0 then b else assoc_pair x l

let rec subst_meta_instances bl c =
  match kind_of_term c with
    | Meta i -> (try assoc_pair i bl with Not_found -> c)
    | _ -> map_constr (subst_meta_instances bl) c

let pose_all_metas_as_evars env evd t =
  let evdref = ref evd in
  let rec aux t = match kind_of_term t with
  | Meta mv ->
      (match Evd.meta_opt_fvalue !evdref mv with
       | Some ({rebus=c},_) -> c
       | None ->
        let {rebus=ty;freemetas=mvs} = Evd.meta_ftype evd mv in
        let ty = if mvs = Evd.Metaset.empty then ty else aux ty in
        let ev = Evarutil.e_new_evar evdref env ~src:(dummy_loc,GoalEvar) ty in
        evdref := meta_assign mv (ev,(ConvUpToEta 0,TypeNotProcessed)) !evdref;
        ev)
  | _ ->
      map_constr aux t in
  let c = aux t in
  (* side-effect *)
  (!evdref, c)

let solve_pattern_eqn_array (env,nb) f l c (sigma,metasubst,evarsubst) =
  match kind_of_term f with
    | Meta k ->
	let c = solve_pattern_eqn env (Array.to_list l) c in
	let n = Array.length l - List.length (fst (decompose_lam c)) in
	let pb = (ConvUpToEta n,TypeNotProcessed) in
	  if noccur_between 1 nb c then
            sigma,(k,lift (-nb) c,pb)::metasubst,evarsubst
	  else error_cannot_unify_local env sigma (mkApp (f, l),c,c)
    | Evar ev ->
	let sigma,c = pose_all_metas_as_evars env sigma c in
	sigma,metasubst,(env,ev,solve_pattern_eqn env (Array.to_list l) c)::evarsubst
    | _ -> assert false

let push d (env,n) = (push_rel_assum d env,n+1)

(*******************************)

(* Unification à l'ordre 0 de m et n: [unify_0 env sigma cv_pb m n]
   renvoie deux listes:

   metasubst:(int*constr)list    récolte les instances des (Meta k)
   evarsubst:(constr*constr)list récolte les instances des (Const "?k")

   Attention : pas d'unification entre les différences instances d'une
   même meta ou evar, il peut rester des doublons *)

(* Unification order: *)
(* Left to right: unifies first argument and then the other arguments *)
(*let unify_l2r x = List.rev x
(* Right to left: unifies last argument and then the other arguments *)
let unify_r2l x = x

let sort_eqns = unify_r2l
*)

(* Option introduced and activated in Coq 8.3 *)
let global_evars_pattern_unification_flag = ref true

open Goptions
let _ =
  declare_bool_option
    { optsync  = true;
      optname  = "pattern-unification for existential variables in tactics";
      optkey   = ["Tactic";"Evars";"Pattern";"Unification"];
      optread  = (fun () -> !global_evars_pattern_unification_flag);
      optwrite = (:=) global_evars_pattern_unification_flag }

type unify_flags = {
  modulo_conv_on_closed_terms : Names.transparent_state option;
  use_metas_eagerly : bool;
  modulo_delta : Names.transparent_state;
  resolve_evars : bool;
  use_evars_pattern_unification : bool;
  modulo_eta : bool
}

let default_unify_flags = {
  modulo_conv_on_closed_terms = Some full_transparent_state;
  use_metas_eagerly = true;
  modulo_delta = full_transparent_state;
  resolve_evars = false;
  use_evars_pattern_unification = true;
  modulo_eta = true
}

let default_no_delta_unify_flags = {
  modulo_conv_on_closed_terms = Some full_transparent_state;
  use_metas_eagerly = true;
  modulo_delta = empty_transparent_state;
  resolve_evars = false;
  use_evars_pattern_unification = false;
  modulo_eta = true
}

let use_evars_pattern_unification flags =
  !global_evars_pattern_unification_flag && flags.use_evars_pattern_unification
  && Flags.version_strictly_greater Flags.V8_2

let expand_key env = function
  | Some (ConstKey cst) -> constant_opt_value env cst
  | Some (VarKey id) -> (try named_body id env with Not_found -> None)
  | Some (RelKey _) -> None
  | None -> None

let key_of flags f =
  match kind_of_term f with
  | Const cst when is_transparent (ConstKey cst) &&
        Cpred.mem cst (snd flags.modulo_delta) ->
      Some (ConstKey cst)
  | Var id when is_transparent (VarKey id) &&
        Idpred.mem id (fst flags.modulo_delta) ->
      Some (VarKey id)
  | _ -> None

let oracle_order env cf1 cf2 =
  match cf1 with
  | None ->
      (match cf2 with
      | None -> None
      | Some k2 -> Some false)
  | Some k1 ->
      match cf2 with
      | None -> Some true
      | Some k2 -> Some (Conv_oracle.oracle_order k1 k2)

let unify_0_with_initial_metas (sigma,ms,es as subst) conv_at_top env cv_pb flags m n =
  let rec unirec_rec (curenv,nb as curenvnb) pb b ((sigma,metasubst,evarsubst) as substn) curm curn =
    let cM = Evarutil.whd_head_evar sigma curm
    and cN = Evarutil.whd_head_evar sigma curn in
      match (kind_of_term cM,kind_of_term cN) with
	| Meta k1, Meta k2 ->
	    let stM,stN = extract_instance_status pb in
	    if k2 < k1
	    then sigma,(k1,cN,stN)::metasubst,evarsubst
	    else if k1 = k2 then substn
	    else sigma,(k2,cM,stM)::metasubst,evarsubst
	| Meta k, _ when not (dependent cM cN) ->
	    (* Here we check that [cN] does not contain any local variables *)
	    if nb = 0 then
              sigma,(k,cN,snd (extract_instance_status pb))::metasubst,evarsubst
            else if noccur_between 1 nb cN then
              (sigma,
	      (k,lift (-nb) cN,snd (extract_instance_status pb))::metasubst,
              evarsubst)
	    else error_cannot_unify_local curenv sigma (m,n,cN)
	| _, Meta k when not (dependent cN cM) ->
	    (* Here we check that [cM] does not contain any local variables *)
	    if nb = 0 then
              (sigma,(k,cM,fst (extract_instance_status pb))::metasubst,evarsubst)
	    else if noccur_between 1 nb cM
	    then
              (sigma,(k,lift (-nb) cM,fst (extract_instance_status pb))::metasubst,
              evarsubst)
	    else error_cannot_unify_local curenv sigma (m,n,cM)
	| Evar ev, _ -> sigma,metasubst,((curenv, ev,cN)::evarsubst)
	| _, Evar ev -> sigma,metasubst,((curenv, ev,cM)::evarsubst)
	| Lambda (na,t1,c1), Lambda (_,t2,c2) ->
	    unirec_rec (push (na,t1) curenvnb) topconv true
	      (unirec_rec curenvnb topconv true substn t1 t2) c1 c2
	| Prod (na,t1,c1), Prod (_,t2,c2) ->
	    unirec_rec (push (na,t1) curenvnb) (prod_pb pb) true
	      (unirec_rec curenvnb topconv true substn t1 t2) c1 c2
	| LetIn (_,a,_,c), _ -> unirec_rec curenvnb pb b substn (subst1 a c) cN
	| _, LetIn (_,a,_,c) -> unirec_rec curenvnb pb b substn cM (subst1 a c)

        (* eta-expansion *)
	| Lambda (na,t1,c1), _ when flags.modulo_eta ->
	    unirec_rec (push (na,t1) curenvnb) topconv true substn
	      c1 (mkApp (lift 1 cN,[|mkRel 1|]))
	| _, Lambda (na,t2,c2) when flags.modulo_eta ->
	    unirec_rec (push (na,t2) curenvnb) topconv true substn
	      (mkApp (lift 1 cM,[|mkRel 1|])) c2

	| Case (_,p1,c1,cl1), Case (_,p2,c2,cl2) ->
            array_fold_left2 (unirec_rec curenvnb topconv true)
	      (unirec_rec curenvnb topconv true
		  (unirec_rec curenvnb topconv true substn p1 p2) c1 c2) cl1 cl2

	| App (f1,l1), _ when
	    (isMeta f1 || use_evars_pattern_unification flags && isEvar f1) &
	    is_unification_pattern curenvnb f1 l1 cN &
	    not (dependent f1 cN) ->
	      solve_pattern_eqn_array curenvnb f1 l1 cN substn

	| _, App (f2,l2) when
	    (isMeta f2 || use_evars_pattern_unification flags && isEvar f2) &
	    is_unification_pattern curenvnb f2 l2 cM &
	    not (dependent f2 cM) ->
	      solve_pattern_eqn_array curenvnb f2 l2 cM substn

	| App (f1,l1), App (f2,l2) ->
	    let len1 = Array.length l1
	    and len2 = Array.length l2 in
	      (try
		  let (f1,l1,f2,l2) =
		    if len1 = len2 then (f1,l1,f2,l2)
		    else if len1 < len2 then
		      let extras,restl2 = array_chop (len2-len1) l2 in
		      (f1, l1, appvect (f2,extras), restl2)
		    else
		      let extras,restl1 = array_chop (len1-len2) l1 in
		      (appvect (f1,extras), restl1, f2, l2) in
		  let pb = ConvUnderApp (len1,len2) in
		  array_fold_left2 (unirec_rec curenvnb topconv true)
		    (unirec_rec curenvnb pb true substn f1 f2) l1 l2
		with ex when precatchable_exception ex ->
		  try expand curenvnb pb b substn cM f1 l1 cN f2 l2
		  with ex when precatchable_exception ex ->
		    canonical_projections curenvnb pb b cM cN substn)

	| _ ->
	    try canonical_projections curenvnb pb b cM cN substn
	    with ex when precatchable_exception ex ->
              if constr_cmp (conv_pb_of cv_pb) cM cN then substn else
		let (f1,l1) =
		  match kind_of_term cM with App (f,l) -> (f,l) | _ -> (cM,[||]) in
		let (f2,l2) =
		  match kind_of_term cN with App (f,l) -> (f,l) | _ -> (cN,[||]) in
		  expand curenvnb pb b substn cM f1 l1 cN f2 l2

  and expand (curenv,_ as curenvnb) pb b (sigma,metasubst,_ as substn) cM f1 l1 cN f2 l2 =

    if
      (* Try full conversion on meta-free terms. *)
      (* Back to 1995 (later on called trivial_unify in 2002), the
	 heuristic was to apply conversion on meta-free (but not
	 evar-free!) terms in all cases (i.e. for apply but also for
	 auto and rewrite, even though auto and rewrite did not use
	 modulo conversion in the rest of the unification
	 algorithm). By compatibility we need to support this
	 separately from the main unification algorithm *)
      (* The exploitation of known metas has been added in May 2007
	 (it is used by apply and rewrite); it might now be redundant
	 with the support for delta-expansion (which is used
	 essentially for apply)... *)
      match flags.modulo_conv_on_closed_terms with
      | None -> false
      | Some convflags ->
      let subst = if flags.use_metas_eagerly then metasubst else ms in
      match subst_defined_metas subst cM with
      | None -> (* some undefined Metas in cM *) false
      | Some m1 ->
      match subst_defined_metas subst cN with
      | None -> (* some undefined Metas in cN *) false
      | Some n1 ->
	  if is_trans_fconv (conv_pb_of pb) convflags env sigma m1 n1
	  then true else
	    if is_ground_term sigma m1 && is_ground_term sigma n1 then
	      error_cannot_unify curenv sigma (cM,cN)
	    else false
    then
      substn
    else
      if b then
      (* Try delta-expansion if in subterms or if asked to conv at top *)
      let cf1 = key_of flags f1 and cf2 = key_of flags f2 in
	match oracle_order curenv cf1 cf2 with
	| None -> error_cannot_unify curenv sigma (cM,cN)
	| Some true ->
	    (match expand_key curenv cf1 with
	    | Some c ->
		unirec_rec curenvnb pb b substn
                  (whd_betaiotazeta sigma (mkApp(c,l1))) cN
	    | None ->
		(match expand_key curenv cf2 with
		| Some c ->
		    unirec_rec curenvnb pb b substn cM
                      (whd_betaiotazeta sigma (mkApp(c,l2)))
		| None ->
		    error_cannot_unify curenv sigma (cM,cN)))
	| Some false ->
	    (match expand_key curenv cf2 with
	    | Some c ->
		unirec_rec curenvnb pb b substn cM
                  (whd_betaiotazeta sigma (mkApp(c,l2)))
	    | None ->
		(match expand_key curenv cf1 with
		| Some c ->
		    unirec_rec curenvnb pb b substn
                      (whd_betaiotazeta sigma (mkApp(c,l1))) cN
		| None ->
		    error_cannot_unify curenv sigma (cM,cN)))
    else
      error_cannot_unify curenv sigma (cM,cN)

  and canonical_projections curenvnb pb b cM cN (sigma,_,_ as substn) =
    let f1 () =
      if isApp cM then
	let f1l1 = decompose_app cM in
	  if is_open_canonical_projection sigma f1l1 then
	    let f2l2 = decompose_app cN in
	      solve_canonical_projection curenvnb pb b cM f1l1 cN f2l2 substn
	  else error_cannot_unify (fst curenvnb) sigma (cM,cN)
      else error_cannot_unify (fst curenvnb) sigma (cM,cN)
    in
      if flags.modulo_conv_on_closed_terms = None then
	error_cannot_unify (fst curenvnb) sigma (cM,cN)
      else
	try f1 () with e when precatchable_exception e ->
	  if isApp cN then
	    let f2l2 = decompose_app cN in
	      if is_open_canonical_projection sigma f2l2 then
		let f1l1 = decompose_app cM in
		  solve_canonical_projection curenvnb pb b cN f2l2 cM f1l1 substn
	      else error_cannot_unify (fst curenvnb) sigma (cM,cN)
	  else error_cannot_unify (fst curenvnb) sigma (cM,cN)

  and solve_canonical_projection curenvnb pb b cM f1l1 cN f2l2 (sigma,ms,es) =
    let (c,bs,(params,params1),(us,us2),(ts,ts1),c1,(n,t2)) =
      try Evarconv.check_conv_record f1l1 f2l2
      with Not_found -> error_cannot_unify (fst curenvnb) sigma (cM,cN)
    in
    let (evd,ks,_) =
      List.fold_left
	(fun (evd,ks,m) b ->
	  if m=n then (evd,t2::ks, m-1) else
            let mv = new_meta () in
	    let evd' = meta_declare mv (substl ks b) evd in
	      (evd', mkMeta mv :: ks, m - 1))
	(sigma,[],List.length bs - 1) bs
    in
    let unilist2 f substn l l' =
      try List.fold_left2 f substn l l'
      with Invalid_argument "List.fold_left2" -> error_cannot_unify (fst curenvnb) sigma (cM,cN)
    in
    let substn = unilist2 (fun s u1 u -> unirec_rec curenvnb pb b s u1 (substl ks u))
      (evd,ms,es) us2 us in
    let substn = unilist2 (fun s u1 u -> unirec_rec curenvnb pb b s u1 (substl ks u))
      substn params1 params in
    let substn = unilist2 (unirec_rec curenvnb pb b) substn ts ts1 in
      unirec_rec curenvnb pb b substn c1 (applist (c,(List.rev ks)))

  in
  let evd = create_evar_defs sigma in
    if (if occur_meta_or_undefined_evar evd m || occur_meta_or_undefined_evar evd n then false 
      else if (match flags.modulo_conv_on_closed_terms with
      | Some flags ->
	  is_trans_fconv (conv_pb_of cv_pb) flags env sigma m n
      | None -> constr_cmp (conv_pb_of cv_pb) m n) then true 
      else if (match flags.modulo_conv_on_closed_terms, flags.modulo_delta with
            | Some (cv_id, cv_k), (dl_id, dl_k) ->
                Idpred.subset dl_id cv_id && Cpred.subset dl_k cv_k
            | None,(dl_id, dl_k) ->
                Idpred.is_empty dl_id && Cpred.is_empty dl_k)
      then error_cannot_unify env sigma (m, n) else false)
    then subst
    else unirec_rec (env,0) cv_pb conv_at_top subst m n

let unify_0 env sigma = unify_0_with_initial_metas (sigma,[],[]) true env

let left = true
let right = false

let pop k = if k=0 then 0 else k-1

let rec unify_with_eta keptside flags env sigma k1 k2 c1 c2 =
  (* Reason up to limited eta-expansion: ci is allowed to start with ki lam *)
  (* Question: try whd_betadeltaiota on ci if ki>0 ? *)
  match kind_of_term c1, kind_of_term c2 with
  | (Lambda (na,t1,c1'), Lambda (_,t2,c2')) ->
      let env' = push_rel_assum (na,t1) env in
      let sigma,metas,evars = unify_0 env sigma topconv flags t1 t2 in
      let side,status,(sigma,metas',evars') =
	unify_with_eta keptside flags env' sigma (pop k1) (pop k2) c1' c2'
      in (side,status,(sigma,metas@metas',evars@evars'))
  | (Lambda (na,t,c1'),_) when k2 > 0 ->
      let env' = push_rel_assum (na,t) env in
      let side = left in (* expansion on the right: we keep the left side *)
      unify_with_eta side flags env' sigma (pop k1) (k2-1)
	c1' (mkApp (lift 1 c2,[|mkRel 1|]))
  | (_,Lambda (na,t,c2')) when k1 > 0 ->
      let env' = push_rel_assum (na,t) env in
      let side = right in (* expansion on the left: we keep the right side *)
      unify_with_eta side flags env' sigma (k1-1) (pop k2)
	(mkApp (lift 1 c1,[|mkRel 1|])) c2'
  | _ ->
      (keptside,ConvUpToEta(min k1 k2),
       unify_0 env sigma topconv flags c1 c2)

(* We solved problems [?n =_pb u] (i.e. [u =_(opp pb) ?n]) and [?n =_pb' u'],
   we now compute the problem on [u =? u'] and decide which of u or u' is kept

   Rem: the upper constraint is lost in case u <= ?n <= u' (and symmetrically
   in the case u' <= ?n <= u)
 *)

let merge_instances env sigma flags st1 st2 c1 c2 =
  match (opp_status st1, st2) with
  | (UserGiven, ConvUpToEta n2) ->
      unify_with_eta left flags env sigma 0 n2 c1 c2
  | (ConvUpToEta n1, UserGiven) ->
      unify_with_eta right flags env sigma n1 0 c1 c2
  | (ConvUpToEta n1, ConvUpToEta n2) ->
      let side = left (* arbitrary choice, but agrees with compatibility *) in
      unify_with_eta side flags env sigma n1 n2 c1 c2
  | ((IsSubType | ConvUpToEta _ | UserGiven as oppst1),
     (IsSubType | ConvUpToEta _ | UserGiven)) ->
      let res = unify_0 env sigma Cumul flags c2 c1 in
      if oppst1=st2 then (* arbitrary choice *) (left, st1, res)
      else if st2=IsSubType or st1=UserGiven then (left, st1, res)
      else (right, st2, res)
  | ((IsSuperType | ConvUpToEta _ | UserGiven as oppst1),
     (IsSuperType | ConvUpToEta _ | UserGiven)) ->
      let res = unify_0 env sigma Cumul flags c1 c2 in
      if oppst1=st2 then (* arbitrary choice *) (left, st1, res)
      else if st2=IsSuperType or st1=UserGiven then (left, st1, res)
      else (right, st2, res)
  | (IsSuperType,IsSubType) ->
      (try (left, IsSubType, unify_0 env sigma Cumul flags c2 c1)
       with _ -> (right, IsSubType, unify_0 env sigma Cumul flags c1 c2))
  | (IsSubType,IsSuperType) ->
      (try (left, IsSuperType, unify_0 env sigma Cumul flags c1 c2)
       with _ -> (right, IsSuperType, unify_0 env sigma Cumul flags c2 c1))

(* Unification
 *
 * Procedure:
 * (1) The function [unify mc wc M N] produces two lists:
 *     (a) a list of bindings Meta->RHS
 *     (b) a list of bindings EVAR->RHS
 *
 * The Meta->RHS bindings cannot themselves contain
 * meta-vars, so they get applied eagerly to the other
 * bindings.  This may or may not close off all RHSs of
 * the EVARs.  For each EVAR whose RHS is closed off,
 * we can just apply it, and go on.  For each which
 * is not closed off, we need to do a mimick step -
 * in general, we have something like:
 *
 *      ?X == (c e1 e2 ... ei[Meta(k)] ... en)
 *
 * so we need to do a mimick step, converting ?X
 * into
 *
 *      ?X -> (c ?z1 ... ?zn)
 *
 * of the proper types.  Then, we can decompose the
 * equation into
 *
 *      ?z1 --> e1
 *          ...
 *      ?zi --> ei[Meta(k)]
 *          ...
 *      ?zn --> en
 *
 * and keep on going.  Whenever we find that a R.H.S.
 * is closed, we can, as before, apply the constraint
 * directly.  Whenever we find an equation of the form:
 *
 *      ?z -> Meta(n)
 *
 * we can reverse the equation, put it into our metavar
 * substitution, and keep going.
 *
 * The most efficient mimick possible is, for each
 * Meta-var remaining in the term, to declare a
 * new EVAR of the same type.  This is supposedly
 * determinable from the clausale form context -
 * we look up the metavar, take its type there,
 * and apply the metavar substitution to it, to
 * close it off.  But this might not always work,
 * since other metavars might also need to be resolved. *)

let applyHead env evd n c  =
  let rec apprec n c cty evd =
    if n = 0 then
      (evd, c)
    else
      match kind_of_term (whd_betadeltaiota env evd cty) with
        | Prod (_,c1,c2) ->
            let (evd',evar) =
	      Evarutil.new_evar evd env ~src:(dummy_loc,GoalEvar) c1 in
	    apprec (n-1) (mkApp(c,[|evar|])) (subst1 evar c2) evd'
	| _ -> error "Apply_Head_Then"
  in
  apprec n c (Typing.type_of env evd c) evd

let is_mimick_head f =
  match kind_of_term f with
      (Const _|Var _|Rel _|Construct _|Ind _) -> true
    | _ -> false

let try_to_coerce env evd c cty tycon =
  let j = make_judge c cty in
  let (evd',j') = inh_conv_coerce_rigid_to dummy_loc env evd j tycon in
  let evd' = Evarconv.consider_remaining_unif_problems env evd' in
  let evd' = Evd.map_metas_fvalue (nf_evar evd') evd' in
    (evd',j'.uj_val)

let w_coerce_to_type env evd c cty mvty =
  let evd,mvty = pose_all_metas_as_evars env evd mvty in
  let tycon = mk_tycon_type mvty in
  try try_to_coerce env evd c cty tycon
  with e when precatchable_exception e ->
    (* inh_conv_coerce_rigid_to should have reasoned modulo reduction
       but there are cases where it though it was not rigid (like in
       fst (nat,nat)) and stops while it could have seen that it is rigid *)
    let cty = Tacred.hnf_constr env evd cty in
    try_to_coerce env evd c cty tycon

let w_coerce env evd mv c =
  let cty = get_type_of env evd c in
  let mvty = Typing.meta_type evd mv in
  w_coerce_to_type env evd c cty mvty

let unify_to_type env sigma flags c status u =
  let c = refresh_universes c in
  let t = get_type_of env sigma c in
  let t = Tacred.hnf_constr env sigma (nf_betaiota sigma (nf_meta sigma t)) in
  let u = Tacred.hnf_constr env sigma u in
    try
      if status = IsSuperType then
	unify_0 env sigma Cumul flags u t
      else if status = IsSubType then
	unify_0 env sigma Cumul flags t u
      else
	try unify_0 env sigma Cumul flags t u
	with e when precatchable_exception e ->
	  unify_0 env sigma Cumul flags u t
    with e when precatchable_exception e ->
      (sigma,[],[])

let unify_type env sigma flags mv status c =
  let mvty = Typing.meta_type sigma mv in
  if occur_meta_or_existential mvty or is_arity env sigma mvty then
    unify_to_type env sigma flags c status mvty
  else (sigma,[],[])

(* Move metas that may need coercion at the end of the list of instances *)

let order_metas metas =
  let rec order latemetas = function
  | [] -> List.rev latemetas
  | (_,_,(status,to_type) as meta)::metas ->
      if to_type = CoerceToType then order (meta::latemetas) metas
      else meta :: order latemetas metas
  in order [] metas

(* Solve an equation ?n[x1=u1..xn=un] = t where ?n is an evar *)

let solve_simple_evar_eqn ts env evd ev rhs =
  let evd,b = solve_simple_eqn (Evarconv.evar_conv_x ts) env evd (None,ev,rhs) in
  if not b then error_cannot_unify env evd (mkEvar ev,rhs);
  Evarconv.consider_remaining_unif_problems env evd

(* [w_merge env sigma b metas evars] merges common instances in metas
   or in evars, possibly generating new unification problems; if [b]
   is true, unification of types of metas is required *)

let w_merge env with_types flags (evd,metas,evars) =
  let rec w_merge_rec evd metas evars eqns =

    (* Process evars *)
    match evars with
    | (curenv,(evk,_ as ev),rhs)::evars' ->
	if Evd.is_defined evd evk then
	  let v = Evd.existential_value evd ev in
	  let (evd,metas',evars'') =
	    unify_0 curenv evd topconv flags rhs v in
	  w_merge_rec evd (metas'@metas) (evars''@evars') eqns
    	else begin
          let rhs' = subst_meta_instances metas rhs in
          match kind_of_term rhs with
	  | App (f,cl) when occur_meta rhs' ->
	      if occur_evar evk rhs' then
                error_occur_check curenv evd evk rhs';
	      if is_mimick_head f then
		let evd' =
		  mimick_undefined_evar evd flags f (Array.length cl) evk in
		w_merge_rec evd' metas evars eqns
	      else
		let evd', rhs'' = pose_all_metas_as_evars curenv evd rhs' in
		w_merge_rec (solve_simple_evar_eqn flags.modulo_delta curenv evd' ev rhs'')
		  metas evars' eqns

          | _ ->
	    let evd', rhs'' = pose_all_metas_as_evars curenv evd rhs' in
	      w_merge_rec (solve_simple_evar_eqn flags.modulo_delta curenv evd' ev rhs'')
		metas evars' eqns
	end
    | [] ->

    (* Process metas *)
    match metas with
    | (mv,c,(status,to_type))::metas ->
        let ((evd,c),(metas'',evars'')),eqns =
	  if with_types & to_type <> TypeProcessed then
	    if to_type = CoerceToType then
              (* Some coercion may have to be inserted *)
	      (w_coerce env evd mv c,([],[])),eqns
	    else
              (* No coercion needed: delay the unification of types *)
	      ((evd,c),([],[])),(mv,status,c)::eqns
	  else
	    ((evd,c),([],[])),eqns in
	  if meta_defined evd mv then
	    let {rebus=c'},(status',_) = meta_fvalue evd mv in
            let (take_left,st,(evd,metas',evars')) =
	      merge_instances env evd flags status' status c' c
	    in
	    let evd' =
              if take_left then evd
              else meta_reassign mv (c,(st,TypeProcessed)) evd
	    in
              w_merge_rec evd' (metas'@metas@metas'') (evars'@evars'') eqns
    	  else
	    let evd' = meta_assign mv (c,(status,TypeProcessed)) evd in
	      w_merge_rec evd' (metas@metas'') evars'' eqns
    | [] ->

    (* Process type eqns *)
    match eqns with
    | (mv,status,c)::eqns ->
        let (evd,metas,evars) = unify_type env evd flags mv status c in
        w_merge_rec evd metas evars eqns
    | [] -> evd

  and mimick_undefined_evar evd flags hdc nargs sp =
    let ev = Evd.find_undefined evd sp in
    let sp_env = Global.env_of_context ev.evar_hyps in
    let (evd', c) = applyHead sp_env evd nargs hdc in
    let (evd'',mc,ec) =
      unify_0 sp_env evd' Cumul flags
        (Retyping.get_type_of sp_env evd' c) ev.evar_concl in
    let evd''' = w_merge_rec evd'' mc ec [] in
    if evd' == evd'''
    then Evd.define sp c evd'''
    else Evd.define sp (Evarutil.nf_evar evd''' c) evd''' in

  (* merge constraints *)
  w_merge_rec evd (order_metas metas) evars []

let w_unify_meta_types env ?(flags=default_unify_flags) evd =
  let metas,evd = retract_coercible_metas evd in
  w_merge env true flags (evd,metas,[])

(* [w_unify env evd M N]
   performs a unification of M and N, generating a bunch of
   unification constraints in the process.  These constraints
   are processed, one-by-one - they may either generate new
   bindings, or, if there is already a binding, new unifications,
   which themselves generate new constraints.  This continues
   until we get failure, or we run out of constraints.
   [clenv_typed_unify M N clenv] expects in addition that expected
   types of metavars are unifiable with the types of their instances    *)

let check_types env flags (sigma,_,_ as subst) m n =
  if isEvar_or_Meta (fst (whd_stack sigma m)) then
    unify_0_with_initial_metas subst true env Cumul
      flags
      (Retyping.get_type_of env sigma n)
      (Retyping.get_type_of env sigma m)
  else if isEvar_or_Meta (fst (whd_stack sigma n)) then
    unify_0_with_initial_metas subst true env Cumul
      flags
      (Retyping.get_type_of env sigma m)
      (Retyping.get_type_of env sigma n)
  else subst

let w_unify_core_0 env with_types cv_pb flags m n evd =
  let (mc1,evd') = retract_coercible_metas evd in
  let (sigma,ms,es) = check_types env flags (evd,mc1,[]) m n in
  let subst2 =
     unify_0_with_initial_metas (evd',ms,es) true env cv_pb flags m n
  in
  let evd = w_merge env with_types flags subst2 in
    if flags.resolve_evars then
      try Typeclasses.resolve_typeclasses ~onlyargs:false ~split:false
	~fail:true env evd
      with e when Typeclasses_errors.unsatisfiable_exception e ->
	error_cannot_unify env evd (m, n)
    else evd

let w_unify_0 env = w_unify_core_0 env false
let w_typed_unify env = w_unify_core_0 env true

(* takes a substitution s, an open term op and a closed term cl
   try to find a subterm of cl which matches op, if op is just a Meta
   FAIL because we cannot find a binding *)

let iter_fail f a =
  let n = Array.length a in
  let rec ffail i =
    if i = n then error "iter_fail"
    else
      try f a.(i)
      with ex when precatchable_exception ex -> ffail (i+1)
  in ffail 0

(* Tries to find an instance of term [cl] in term [op].
   Unifies [cl] to every subterm of [op] until it finds a match.
   Fails if no match is found *)
let w_unify_to_subterm env ?(flags=default_unify_flags) (op,cl) evd =
  let rec matchrec cl =
    let cl = strip_outer_cast cl in
    (try
       if closed0 cl && not (isEvar cl)
       then w_typed_unify env topconv flags op cl evd,cl
       else error "Bound 1"
     with ex when precatchable_exception ex ->
       (match kind_of_term cl with
	  | App (f,args) ->
	      let n = Array.length args in
	      assert (n>0);
	      let c1 = mkApp (f,Array.sub args 0 (n-1)) in
	      let c2 = args.(n-1) in
	      (try
		 matchrec c1
	       with ex when precatchable_exception ex ->
		 matchrec c2)
          | Case(_,_,c,lf) -> (* does not search in the predicate *)
	       (try
		 matchrec c
	       with ex when precatchable_exception ex ->
		 iter_fail matchrec lf)
	  | LetIn(_,c1,_,c2) ->
	       (try
		 matchrec c1
	       with ex when precatchable_exception ex ->
		 matchrec c2)

	  | Fix(_,(_,types,terms)) ->
	       (try
		 iter_fail matchrec types
	       with ex when precatchable_exception ex ->
		 iter_fail matchrec terms)

	  | CoFix(_,(_,types,terms)) ->
	       (try
		 iter_fail matchrec types
	       with ex when precatchable_exception ex ->
		 iter_fail matchrec terms)

          | Prod (_,t,c) ->
	      (try
		 matchrec t
	       with ex when precatchable_exception ex ->
		 matchrec c)
          | Lambda (_,t,c) ->
	      (try
		 matchrec t
	       with ex when precatchable_exception ex ->
		 matchrec c)
          | _ -> error "Match_subterm"))
  in
  try matchrec cl
  with ex when precatchable_exception ex ->
    raise (PretypeError (env,evd,NoOccurrenceFound (op, None)))

(* Tries to find all instances of term [cl] in term [op].
   Unifies [cl] to every subterm of [op] and return all the matches.
   Fails if no match is found *)
let w_unify_to_subterm_all env ?(flags=default_unify_flags) (op,cl) evd =
  let return a b =
    let (evd,c as a) = a () in
      if List.exists (fun (evd',c') -> eq_constr c c') b then b else a :: b
  in
  let fail str _ = error str in
  let bind f g a =
    let a1 = try f a
             with ex
             when precatchable_exception ex -> a
    in try g a1
       with ex
       when precatchable_exception ex -> a1
  in
  let bind_iter f a =
    let n = Array.length a in
    let rec ffail i =
      if i = n then fun a -> a
      else bind (f a.(i)) (ffail (i+1))
    in ffail 0
  in
  let rec matchrec cl =
    let cl = strip_outer_cast cl in
      (bind
	  (if closed0 cl
	  then return (fun () -> w_typed_unify env topconv flags op cl evd,cl)
            else fail "Bound 1")
          (match kind_of_term cl with
	    | App (f,args) ->
		let n = Array.length args in
		assert (n>0);
		let c1 = mkApp (f,Array.sub args 0 (n-1)) in
		let c2 = args.(n-1) in
		bind (matchrec c1) (matchrec c2)

            | Case(_,_,c,lf) -> (* does not search in the predicate *)
		bind (matchrec c) (bind_iter matchrec lf)

	    | LetIn(_,c1,_,c2) ->
		bind (matchrec c1) (matchrec c2)

	    | Fix(_,(_,types,terms)) ->
		bind (bind_iter matchrec types) (bind_iter matchrec terms)

	    | CoFix(_,(_,types,terms)) ->
		bind (bind_iter matchrec types) (bind_iter matchrec terms)

            | Prod (_,t,c) ->
		bind (matchrec t) (matchrec c)

            | Lambda (_,t,c) ->
		bind (matchrec t) (matchrec c)

            | _ -> fail "Match_subterm"))
  in
  let res = matchrec cl [] in
  if res = [] then
    raise (PretypeError (env,evd,NoOccurrenceFound (op, None)))
  else
    res

let w_unify_to_subterm_list env flags allow_K hdmeta oplist t evd =
  List.fold_right
    (fun op (evd,l) ->
      let op = whd_meta evd op in
      if isMeta op then
	if allow_K then (evd,op::l)
	else error_abstraction_over_meta env evd hdmeta (destMeta op)
      else if occur_meta_or_existential op then
        let (evd',cl) =
          try
	    (* This is up to delta for subterms w/o metas ... *)
	    w_unify_to_subterm env ~flags (strip_outer_cast op,t) evd
	  with PretypeError (env,_,NoOccurrenceFound _) when allow_K -> (evd,op)
        in
	  if not allow_K && (* ensure we found a different instance *)
	    List.exists (fun op -> eq_constr op cl) l
	  then error_non_linear_unification env evd hdmeta cl
	  else (evd',cl::l)
      else if allow_K or dependent op t then
	(evd,op::l)
      else
	(* This is not up to delta ... *)
	raise (PretypeError (env,evd,NoOccurrenceFound (op, None))))
    oplist
    (evd,[])

let secondOrderAbstraction env flags allow_K typ (p, oplist) evd =
  (* Remove delta when looking for a subterm *)
  let flags = { flags with modulo_delta = (fst flags.modulo_delta, Cpred.empty) } in
  let (evd',cllist) =
    w_unify_to_subterm_list env flags allow_K p oplist typ evd in
  let typp = Typing.meta_type evd' p in
  let pred = abstract_list_all env evd' typp typ cllist in
  w_merge env false flags (evd',[p,pred,(ConvUpToEta 0,TypeProcessed)],[])

let w_unify2 env flags allow_K cv_pb ty1 ty2 evd =
  let c1, oplist1 = whd_stack evd ty1 in
  let c2, oplist2 = whd_stack evd ty2 in
  match kind_of_term c1, kind_of_term c2 with
    | Meta p1, _ ->
        (* Find the predicate *)
	let evd' =
          secondOrderAbstraction env flags allow_K ty2 (p1,oplist1) evd in
        (* Resume first order unification *)
	w_unify_0 env cv_pb flags (nf_meta evd' ty1) ty2 evd'
    | _, Meta p2 ->
        (* Find the predicate *)
	let evd' =
          secondOrderAbstraction env flags allow_K ty1 (p2, oplist2) evd in
        (* Resume first order unification *)
	w_unify_0 env cv_pb flags ty1 (nf_meta evd' ty2) evd'
    | _ -> error "w_unify2"

(* The unique unification algorithm works like this: If the pattern is
   flexible, and the goal has a lambda-abstraction at the head, then
   we do a first-order unification.

   If the pattern is not flexible, then we do a first-order
   unification, too.

   If the pattern is flexible, and the goal doesn't have a
   lambda-abstraction head, then we second-order unification. *)

(* We decide here if first-order or second-order unif is used for Apply *)
(* We apply a term of type (ai:Ai)C and try to solve a goal C'          *)
(* The type C is in clenv.templtyp.rebus with a lot of Meta to solve    *)

(* 3-4-99 [HH] New fo/so choice heuristic :
   In case we have to unify (Meta(1) args) with ([x:A]t args')
   we first try second-order unification and if it fails first-order.
   Before, second-order was used if the type of Meta(1) and [x:A]t was
   convertible and first-order otherwise. But if failed if e.g. the type of
   Meta(1) had meta-variables in it. *)
let w_unify allow_K env cv_pb ?(flags=default_unify_flags) ty1 ty2 evd =
  let cv_pb = of_conv_pb cv_pb in
  let hd1,l1 = whd_stack evd ty1 in
  let hd2,l2 = whd_stack evd ty2 in
    match kind_of_term hd1, l1<>[], kind_of_term hd2, l2<>[] with
      (* Pattern case *)
      | (Meta _, true, Lambda _, _ | Lambda _, _, Meta _, true)
	  when List.length l1 = List.length l2 ->
	  (try
	      w_typed_unify env cv_pb flags ty1 ty2 evd
	    with ex when precatchable_exception ex ->
	      try
		w_unify2 env flags allow_K cv_pb ty1 ty2 evd
	      with PretypeError (env,_,NoOccurrenceFound _) as e -> raise e)

      (* Second order case *)
      | (Meta _, true, _, _ | _, _, Meta _, true) ->
	  (try
	      w_unify2 env flags allow_K cv_pb ty1 ty2 evd
	    with PretypeError (env,_,NoOccurrenceFound _) as e -> raise e
	      | ex when precatchable_exception ex ->
		  try
		    w_typed_unify env cv_pb flags ty1 ty2 evd
		  with ex' when precatchable_exception ex' ->
		    raise ex)

      (* General case: try first order *)
      | _ -> w_typed_unify env cv_pb flags ty1 ty2 evd