(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

open Sorts
open Util
open CErrors
open Names
open Context
open Constr
open Environ
open Termops
open Evd
open EConstr
open Vars
open Namegen
open Retyping
open Reductionops
open Evarutil
open Pretype_errors

module AllowedEvars = struct

  type t =
  | AllowAll
  | AllowFun of (Evar.t -> bool) * Evar.Set.t

  let mem allowed evk =
    match allowed with
    | AllowAll -> true
    | AllowFun (f,except) -> f evk && not (Evar.Set.mem evk except)

  let remove evk = function
    | AllowAll -> AllowFun ((fun _ -> true), Evar.Set.singleton evk)
    | AllowFun (f,except) -> AllowFun (f, Evar.Set.add evk except)

  let all = AllowAll

  let except evars =
    AllowFun ((fun _ -> true), evars)

  let from_pred f =
    AllowFun (f, Evar.Set.empty)

end

type unify_flags = {
  modulo_betaiota: bool;
  open_ts : TransparentState.t;
  closed_ts : TransparentState.t;
  subterm_ts : TransparentState.t;
  allowed_evars : AllowedEvars.t;
  with_cs : bool
}

let is_evar_allowed flags evk =
  AllowedEvars.mem flags.allowed_evars evk

type unification_kind =
  | TypeUnification
  | TermUnification

(************************)
(* Unification results  *)
(************************)

type unification_result =
  | Success of evar_map
  | UnifFailure of evar_map * unification_error

let is_success = function Success _ -> true | UnifFailure _ -> false

let test_success unify flags b env evd c c' rhs =
  is_success (unify flags b env evd c c' rhs)

(** A unification function parameterized by:
    - unification flags
    - the kind of unification
    - environment
    - sigma
    - conversion problem
    - the two terms to unify. *)

type unifier = unify_flags -> unification_kind ->
  env -> evar_map -> conv_pb -> constr -> constr -> unification_result

(** A conversion function: parameterized by the kind of unification,
    environment, sigma, conversion problem and the two terms to convert.
    Conversion is not allowed to instantiate evars contrary to unification. *)
type conversion_check = unify_flags -> unification_kind ->
  env ->  evar_map -> conv_pb -> constr -> constr -> bool

let normalize_evar evd ev =
  match EConstr.kind evd (mkEvar ev) with
  | Evar (evk,args) -> (evk,args)
  | _ -> assert false

let get_polymorphic_positions env sigma f =
  let open Declarations in
  match EConstr.kind sigma f with
  | Ind (ind, u) | Construct ((ind, _), u) ->
    let mib,oib = Inductive.lookup_mind_specif env ind in
      (match mib.mind_template with
      | None -> assert false
      | Some templ -> templ.template_param_levels)
  | _ -> assert false

let refresh_universes ?(status=univ_rigid) ?(onlyalg=false) ?(refreshset=false)
                      pbty env evd t =
  let evdref = ref evd in
  (* direction: true for fresh universes lower than the existing ones *)
  let refresh_sort status ~direction s =
    let sigma, l = new_univ_level_variable status !evdref in
    let s' = match ESorts.kind sigma s with
      | QSort (q, _) -> Sorts.qsort q (Univ.Universe.make l)
      | _ -> Sorts.sort_of_univ @@ Univ.Universe.make l
    in
    let s' = ESorts.make s' in
    evdref := sigma;
    let evd =
      if direction then set_leq_sort env !evdref s' s
      else set_leq_sort env !evdref s s'
    in evdref := evd; mkSort s'
  in
  let rec refresh ~onlyalg status ~direction t =
    match EConstr.kind !evdref t with
    | Sort s ->
      begin match ESorts.kind !evdref s with
      | Type u | QSort (_, u) ->
         (* TODO: check if max(l,u) is not ok as well *)
        (match Univ.Universe.level u with
        | None -> refresh_sort status ~direction s
        | Some l ->
           (match Evd.universe_rigidity !evdref l with
            | UnivRigid ->
              if not onlyalg && (not (Univ.Level.is_set l) || (refreshset && not direction))
              then refresh_sort status ~direction s
               else t
            | UnivFlexible alg ->
               (if alg then
                  evdref := Evd.make_nonalgebraic_variable !evdref l);
               t))
      | Set when refreshset && not direction ->
       (* Cannot make a universe "lower" than "Set",
          only refreshing when we want higher universes. *)
       refresh_sort status ~direction s
      | Prop | SProp | Set -> t
      end
    | Prod (na,u,v) ->
       let v' = refresh ~onlyalg status ~direction v in
       if v' == v then t else mkProd (na, u, v')
    | _ -> t
  in
  (* Refresh the types of evars under template polymorphic references *)
  let rec refresh_term_evars ~onevars ~top t =
    match EConstr.kind !evdref t with
    | App (f, args) when Termops.is_template_polymorphic_ind env !evdref f ->
      let pos = get_polymorphic_positions env !evdref f in
        refresh_polymorphic_positions args pos; t
    | App (f, args) when top && isEvar !evdref f ->
       let f' = refresh_term_evars ~onevars:true ~top:false f in
       let args' = Array.map (refresh_term_evars ~onevars ~top:false) args in
       if f' == f && args' == args then t
       else mkApp (f', args')
    | Evar (ev, a) when onevars ->
      let evi = Evd.find_undefined !evdref ev in
      let ty = Evd.evar_concl evi in
      let ty' = refresh ~onlyalg univ_flexible ~direction:true ty in
      if ty == ty' then t
      else (evdref := Evd.downcast ev ty' !evdref; t)
    | Sort s ->
       (match ESorts.kind !evdref s with
        | Type u when not (Univ.Universe.is_levels u) ->
           refresh_sort Evd.univ_flexible ~direction:false s
        | _ -> t)
    | _ -> EConstr.map !evdref (refresh_term_evars ~onevars ~top:false) t
  and refresh_polymorphic_positions args pos =
    let rec aux i = function
      | Some l :: ls ->
        if i < Array.length args then
          ignore(refresh_term_evars ~onevars:true ~top:false args.(i));
        aux (succ i) ls
      | None :: ls ->
        if i < Array.length args then
          ignore(refresh_term_evars ~onevars:false ~top:false args.(i));
        aux (succ i) ls
      | [] -> ()
    in aux 0 pos
  in
  let t' =
    if isArity !evdref t then
      match pbty with
      | None ->
         (* No cumulativity needed, but we still need to refresh the algebraics *)
         refresh ~onlyalg:true univ_flexible ~direction:false t
      | Some direction -> refresh ~onlyalg status ~direction t
    else refresh_term_evars ~onevars:false ~top:true t
  in !evdref, t'

let get_type_of_refresh  ?(polyprop=true) ?(lax=false) env evars t =
  let tty = Retyping.get_type_of env evars t in
  let evars', tty = refresh_universes ~onlyalg:true
    ~status:(Evd.UnivFlexible false) (Some false) env evars tty in
  evars', tty

let add_conv_oriented_pb ?(tail=true) (pbty,env,t1,t2) evd =
  match pbty with
  | Some true -> add_conv_pb ~tail (Conversion.CUMUL,env,t1,t2) evd
  | Some false -> add_conv_pb ~tail (Conversion.CUMUL,env,t2,t1) evd
  | None -> add_conv_pb ~tail (Conversion.CONV,env,t1,t2) evd

(* We retype applications to ensure the universe constraints are collected *)

exception IllTypedInstance of env * evar_map * EConstr.types * EConstr.types
exception IllTypedInstanceFun of env * evar_map * EConstr.constr * EConstr.types

let recheck_applications unify flags env evdref t =
  let rec aux env t =
    match EConstr.kind !evdref t with
    | App (f, args) ->
       let () = aux env f in
       let fty = Retyping.get_type_of env !evdref f in
       let argsty = Array.map (fun x -> aux env x; Retyping.get_type_of env !evdref x) args in
       let rec aux i ty =
         if i < Array.length argsty then
         match EConstr.kind !evdref (whd_all env !evdref ty) with
         | Prod (na, dom, codom) ->
            (match unify flags TypeUnification env !evdref Conversion.CUMUL argsty.(i) dom with
             | Success evd -> evdref := evd;
                             aux (succ i) (subst1 args.(i) codom)
             | UnifFailure (evd, reason) -> raise (IllTypedInstance (env, evd, argsty.(i), dom)))
         | _ -> raise (IllTypedInstanceFun (env, !evdref, f, ty))
       else ()
     in aux 0 fty
    | _ ->
       iter_with_full_binders env !evdref (fun d env -> push_rel d env) aux env t
  in aux env t


(*------------------------------------*
 * Restricting existing evars         *
 *------------------------------------*)

type 'a update =
| UpdateWith of 'a
| NoUpdate

let restrict_evar_key evd evk filter candidates =
  match filter, candidates with
  | None, NoUpdate -> evd, evk
  | _ ->
    let evi = Evd.find_undefined evd evk in
    let oldfilter = evar_filter evi in
    begin match filter, candidates with
    | Some filter, NoUpdate when Filter.equal oldfilter filter ->
      evd, evk
    | _ ->
      let filter = match filter with
      | None -> evar_filter evi
      | Some filter -> filter in
      let candidates = match candidates with
      | NoUpdate -> Evd.evar_candidates evi
      | UpdateWith c -> Some c in
      restrict_evar evd evk filter candidates
    end

(* Restrict an applied evar and returns its restriction in the same context *)
(* (the filter is assumed to be at least stronger than the original one) *)
let restrict_applied_evar evd (evk,argsv) filter candidates =
  let evd,newevk = restrict_evar_key evd evk filter candidates in
  let newargsv = match filter with
  | None -> (* optim *) argsv
  | Some filter ->
      let EvarInfo evi = Evd.find evd evk in
      let subfilter = Filter.compose (evar_filter evi) filter in
      Filter.filter_slist subfilter argsv in
  evd,(newevk,newargsv)

(* Restrict an evar in the current evar_map *)
let restrict_evar evd evk filter candidates =
  fst (restrict_evar_key evd evk filter candidates)

(* Restrict an evar in the current evar_map *)
let restrict_instance evd evk filter argsv =
  match filter with None -> argsv | Some filter ->
  let EvarInfo evi = Evd.find evd evk in
  Filter.filter_slist (Filter.compose (evar_filter evi) filter) argsv

open Context.Rel.Declaration
let noccur_evar env evd evk c =
  let cache = ref Int.Set.empty (* cache for let-ins *) in
  let rec occur_rec check_types (k, env as acc) c =
  match EConstr.kind evd c with
  | Evar (evk',args' as ev') ->
    if Evar.equal evk evk' then raise Occur
    else (if check_types then
            occur_rec false acc (existential_type evd ev');
          SList.Skip.iter (occur_rec check_types acc) args')
  | Rel i when i > k ->
     if not (Int.Set.mem (i-k) !cache) then
       let decl = Environ.lookup_rel i env in
       if check_types then
         (cache := Int.Set.add (i-k) !cache; occur_rec false acc (lift i (EConstr.of_constr (get_type decl))));
       (match decl with
        | LocalAssum _ -> ()
        | LocalDef (_,b,_) -> cache := Int.Set.add (i-k) !cache; occur_rec false acc (lift i (EConstr.of_constr b)))
  | Proj (p,_,c) -> occur_rec true acc c
  | _ -> iter_with_full_binders env evd (fun rd (k,env) -> (succ k, push_rel rd env))
    (occur_rec check_types) acc c
  in
  try occur_rec false (0,env) c; true with Occur -> false

(****************************************)
(* Managing chains of local definitions *)
(****************************************)

type alias =
| RelAlias of int
| VarAlias of Id.t

let of_alias = function
| RelAlias n -> mkRel n
| VarAlias id -> mkVar id

let to_alias sigma c = match EConstr.kind sigma c with
| Rel n -> Some (RelAlias n)
| Var id -> Some (VarAlias id)
| _ -> None

let is_alias sigma c alias = match EConstr.kind sigma c, alias with
| Var id, VarAlias id' -> Id.equal id id'
| Rel n, RelAlias n' -> Int.equal n n'
| _ -> false

let eq_alias a b = match a, b with
| RelAlias n, RelAlias m -> Int.equal m n
| VarAlias id1, VarAlias id2 -> Id.equal id1 id2
| _ -> false

let compare_alias a b = match a, b with
| RelAlias n, RelAlias m -> Int.compare n m
| VarAlias id1, VarAlias id2 -> Id.compare id1 id2
| RelAlias _, VarAlias _ -> -1
| VarAlias _, RelAlias _ -> 1

module AlsOrd = struct type t = alias let compare = compare_alias end
module AlsMap = Map.Make(AlsOrd)

(* A chain of let-in ended either by a declared variable or a non-variable term *)
(* e.g. [x:=t;y:=x;z:=y] binds [z] to [NonVarAliasChain ([y;x],t)] *)
(* and. [a:T;x:=a;y:=x;z:=y] binds [z] to [VarAliasChain ([y;x],a)] *)
type 'a alias_chain =
  | VarAliasChain of alias list * alias
  | NonVarAliasChain of alias list * 'a

let init_var_alias_chain x = VarAliasChain ([], x)
let init_term_alias_chain c = NonVarAliasChain ([], c)

let push_alias aliases_chain a =
  (* most recent variables come first *)
  match aliases_chain with
  | VarAliasChain (l, last) -> VarAliasChain (a :: l, last)
  | NonVarAliasChain (l, last) -> NonVarAliasChain (a :: l, last)

module Alias =
struct
type t = { mutable lift : int; mutable data : EConstr.t }

let make c = { lift = 0; data = c }

let lift n { lift; data } = { lift = lift + n; data }

let eval alias =
  let c = EConstr.Vars.lift alias.lift alias.data in
  let () = alias.lift <- 0 in
  let () = alias.data <- c in
  c

let repr sigma alias = match EConstr.kind sigma alias.data with
| Rel n -> Some (RelAlias (n + alias.lift))
| Var id -> Some (VarAlias id)
| _ -> None

end

let lift_alias_chain n alias_chain =
  let map a = match a with
  | VarAlias _ -> a
  | RelAlias m -> RelAlias (m + n)
  in
  match alias_chain with
  | VarAliasChain (l, alias) -> VarAliasChain (List.map map l, map alias)
  | NonVarAliasChain (l, alias) -> NonVarAliasChain (List.map map l, Alias.lift n alias)

let cast_alias_chain = function
  | VarAliasChain (l, v) -> VarAliasChain (l, v)
  | NonVarAliasChain (l, c) -> NonVarAliasChain (l, Alias.make c)

type aliases = {
  rel_aliases : Alias.t alias_chain Int.Map.t;
  var_aliases : EConstr.t alias_chain Id.Map.t;
  (** Only contains [VarAlias] *)
}

(* Expand rels and vars that are bound to other rels or vars so that
   dependencies in variables are canonically associated to the most ancient
   variable in its family of aliased variables *)

let compute_var_aliases sign sigma =
  let open Context.Named.Declaration in
  (* push from oldest to more recent variables *)
  List.fold_right (fun decl aliases ->
    let id = get_id decl in
    match decl with
    | LocalDef (_,t,_) ->
      let aliases_of_id =
        match EConstr.kind sigma t with
        | Var id' ->
          (try push_alias (Id.Map.find id' aliases) (VarAlias id')
          with Not_found -> init_var_alias_chain (VarAlias id'))
        | _ ->
          init_term_alias_chain t in
      Id.Map.add id aliases_of_id aliases
    | LocalAssum _ -> aliases)
    sign Id.Map.empty

let compute_rel_aliases var_aliases rels sigma =
  (* push from oldest to more recent variables *)
  snd (List.fold_right
         (fun decl (n,aliases) ->
          (n-1,
           match decl with
           | LocalDef (_,t,u) ->
             let aliases_of_n =
               match EConstr.kind sigma t with
               | Var id' ->
                   (let alias = VarAlias id' in
                    try push_alias (cast_alias_chain (Id.Map.find id' var_aliases)) alias
                    with Not_found -> init_var_alias_chain alias)
               | Rel p ->
                   (let alias = RelAlias (p+n) in
                    try push_alias (Int.Map.find (p+n) aliases) alias
                    with Not_found -> init_var_alias_chain alias)
               | _ ->
                  init_term_alias_chain (Alias.lift n (Alias.make @@ mkCast(t,DEFAULTcast, u)))
             in
             Int.Map.add n aliases_of_n aliases
           | LocalAssum _ -> aliases)
         )
         rels
         (List.length rels,Int.Map.empty))

let make_alias_map env sigma =
  (* We compute the chain of aliases for each var and rel *)
  let var_aliases = compute_var_aliases (named_context env) sigma in
  let rel_aliases = compute_rel_aliases var_aliases (rel_context env) sigma in
  { var_aliases; rel_aliases }

let lift_aliases n aliases =
  if Int.equal n 0 then aliases else
  let rel_aliases =
   Int.Map.fold (fun p l -> Int.Map.add (p+n) (lift_alias_chain n l))
     aliases.rel_aliases Int.Map.empty
  in
  { aliases with rel_aliases }

let get_alias_chain_of aliases x = match x with
  | RelAlias n -> (try Some (Int.Map.find n aliases.rel_aliases) with Not_found -> None)
  | VarAlias id -> (try Some (cast_alias_chain (Id.Map.find id aliases.var_aliases)) with Not_found -> None)

(* Expand an alias as much as possible while remaining a variable *)
(* i.e. returns either a declared variable [y], or the last expansion
   of [x] defined to be [c] and [c] is not a variable *)
let normalize_alias aliases x =
  match get_alias_chain_of aliases x with
  | None | Some (NonVarAliasChain ([], _)) -> x
  | Some (NonVarAliasChain (l, _)) -> List.last l
  | Some (VarAliasChain (_, a)) -> a

(* Idem, specifically for named variables *)
let normalize_alias_var var_aliases id =
  let aliases = { var_aliases; rel_aliases = Int.Map.empty } in
  match normalize_alias aliases (VarAlias id) with
  | VarAlias id -> id
  | RelAlias _ -> assert false (** var only aliases to variables *)

let extend_alias sigma decl { var_aliases; rel_aliases } =
  let rel_aliases =
    Int.Map.fold (fun n l -> Int.Map.add (n+1) (lift_alias_chain 1 l))
      rel_aliases Int.Map.empty in
  let rel_aliases =
    match decl with
    | LocalDef(_,t,_) ->
        let aliases_of_binder =
        match EConstr.kind sigma t with
        | Var id' ->
            let alias = VarAlias id' in
            (try push_alias (cast_alias_chain (Id.Map.find id' var_aliases)) alias
            with Not_found -> init_var_alias_chain alias)
        | Rel p ->
            let alias = RelAlias (p+1) in
            (try push_alias (Int.Map.find (p+1) rel_aliases) alias
            with Not_found -> init_var_alias_chain alias)
        | _ ->
           init_term_alias_chain (Alias.lift 1 (Alias.make t))
             in
             Int.Map.add 1 aliases_of_binder rel_aliases
    | LocalAssum _ -> rel_aliases in
  { var_aliases; rel_aliases }

let expand_alias_once aliases x =
  match get_alias_chain_of aliases x with
  | None -> None
  | Some (VarAliasChain (x :: _, _) | NonVarAliasChain (x :: _, _) | VarAliasChain ([], x)) -> Some (Alias.make (of_alias x))
  | Some (NonVarAliasChain ([], a)) -> Some a

let expansions_of_var aliases x =
  match get_alias_chain_of aliases x with
  | None -> [x]
  | Some (VarAliasChain (l, y)) -> x :: l @ [y]
  | Some (NonVarAliasChain (l, _)) -> x :: l

let expansion_of_var sigma aliases x =
  match get_alias_chain_of aliases x with
  | None -> (false, Some x, [])
  | Some (VarAliasChain (l, x)) -> (true, Some x, l)
  | Some (NonVarAliasChain (l, a)) -> (true, Alias.repr sigma a, l)

let rec expand_vars_in_term_using env sigma aliases t = match EConstr.kind sigma t with
  | Rel n -> of_alias (normalize_alias aliases (RelAlias n))
  | Var id -> of_alias (normalize_alias aliases (VarAlias id))
  | _ ->
    let self aliases c = expand_vars_in_term_using env sigma aliases c in
    map_constr_with_full_binders env sigma (extend_alias sigma) self aliases t

let expand_vars_in_term env sigma = expand_vars_in_term_using env sigma (make_alias_map env sigma)

let free_vars_and_rels_up_alias_expansion env sigma aliases c =
  let fv_rels = ref Int.Set.empty and fv_ids = ref Id.Set.empty in
  let let_rels = ref Int.Set.empty and let_ids = ref Id.Set.empty in
  let cache_rel = ref Int.Set.empty and cache_var = ref Id.Set.empty in
  let is_in_cache depth = function
    | RelAlias n -> Int.Set.mem (n-depth) !cache_rel
    | VarAlias s -> Id.Set.mem s !cache_var
  in
  let put_in_cache depth = function
    | RelAlias n -> cache_rel := Int.Set.add (n-depth) !cache_rel
    | VarAlias s -> cache_var := Id.Set.add s !cache_var
  in
  let rec frec (aliases,depth) c =
    match EConstr.kind sigma c with
    | Rel _ | Var _ as ck ->
      let ck = match ck with
      | Rel n -> RelAlias n
      | Var id -> VarAlias id
      | _ -> assert false
      in
      if is_in_cache depth ck then () else begin
      put_in_cache depth ck;
      let expanded, c', l = expansion_of_var sigma aliases ck in
      (if expanded then (* expansion, hence a let-in *)
        List.iter (function
        | VarAlias id -> let_ids := Id.Set.add id !let_ids
        | RelAlias n -> if n >= depth+1 then let_rels := Int.Set.add (n-depth) !let_rels)
       (ck :: l));
      match c' with
        | Some (VarAlias id) -> fv_ids := Id.Set.add id !fv_ids
        | Some (RelAlias n) -> if n >= depth+1 then fv_rels := Int.Set.add (n-depth) !fv_rels
        | None -> frec (aliases,depth) c end
    | Const _ | Ind _ | Construct _ ->
        fv_ids := Id.Set.union (vars_of_global env (fst @@ EConstr.destRef sigma c)) !fv_ids
    | _ ->
        iter_with_full_binders env sigma
          (fun d (aliases,depth) -> (extend_alias sigma d aliases,depth+1))
          frec (aliases,depth) c
  in
  frec (aliases,0) c;
  (!fv_rels,!fv_ids,!let_rels,!let_ids)

(********************************)
(* Managing pattern-unification *)
(********************************)

let expand_and_check_vars aliases l =
  let map a = match get_alias_chain_of aliases a with
  | None -> Some a
  | Some (VarAliasChain (_, a)) -> Some a
  | Some (NonVarAliasChain ([], c)) -> None
  | Some (NonVarAliasChain (l, c)) -> Some (List.last l)
  in
  Option.List.map map l

let alias_distinct l =
  let rec check (rels, vars) = function
  | [] -> true
  | RelAlias n :: l ->
    not (Int.Set.mem n rels) && check (Int.Set.add n rels, vars) l
  | VarAlias id :: l ->
    not (Id.Set.mem id vars) && check (rels, Id.Set.add id vars) l
  in
  check (Int.Set.empty, Id.Set.empty) l

let distinct_actual_deps env evd aliases l t =
  (* If the aliases are already unique, any subset will also be. *)
  if alias_distinct l then true
  (* The instance of a meta can virtually contains any variable of the context *)
  else if occur_meta evd t then false
  else
    (* Probably strong restrictions coming from t being evar-closed *)
    let (fv_rels,fv_ids,_,_) = free_vars_and_rels_up_alias_expansion env evd aliases t in
    alias_distinct @@ List.filter (function
      | VarAlias id -> Id.Set.mem id fv_ids
      | RelAlias n -> Int.Set.mem n fv_rels
    ) l

open Context.Named.Declaration
let remove_instance_local_defs evd evk args =
  let EvarInfo evi = Evd.find evd evk in
  let rec aux sign args = match sign, args with
  | [], [] -> []
  | LocalAssum _ :: sign, c :: args -> c :: aux sign args
  | LocalDef _ :: sign, _ :: args -> aux sign args
  | _ -> assert false
  in
  aux (evar_filtered_context evi) args

(* Check if an applied evar "?X[args] l" is a Miller's pattern *)

let find_unification_pattern_args env evd l t =
  let aliases = make_alias_map env evd in
  match expand_and_check_vars aliases l with
  | Some l as x when distinct_actual_deps env evd aliases l t -> x
  | _ -> None

let is_unification_pattern_meta env evd nb m l t =
  (* Variables from context and rels > nb are implicitly all there *)
  (* so we need to be a rel <= nb *)
  let map a = match EConstr.kind evd a with
  | Rel n -> if n <= nb then Some (RelAlias n) else None
  | _ -> None
  in
  match Option.List.map map l with
  | Some l ->
    begin match find_unification_pattern_args env evd l t with
    | Some _ as x when not (occur_metavariable evd m t) -> x
    | _ -> None
    end
  | None ->
    None

let is_unification_pattern_evar env evd (evk,args) l t =
  match Option.List.map (fun c -> to_alias evd c) l with
  | Some l when noccur_evar env evd evk t ->
    let args = Evd.expand_existential evd (evk, args) in
    let args = remove_instance_local_defs evd evk args in
    let args = Option.List.map (fun c -> to_alias evd c) args in
    begin match args with
    | None -> None
    | Some args ->
    let n = List.length args in
      match find_unification_pattern_args env evd (args @ l) t with
      | Some l -> Some (List.skipn n l)
      | _ -> None
    end
  | _ -> None

let is_unification_pattern_pure_evar env evd (evk,args) t =
  let is_ev = is_unification_pattern_evar env evd (evk,args) [] t in
  match is_ev with
  | None -> false
  | Some _ -> true

let is_unification_pattern (env,nb) evd f l t =
  match EConstr.kind evd f with
  | Meta m -> is_unification_pattern_meta env evd nb m l t
  | Evar ev -> is_unification_pattern_evar env evd ev l t
  | _ -> None

(* From a unification problem "?X l = c", build "\x1...xn.(term1 l2)"
   (pattern unification). It is assumed that l is made of rel's that
   are distinct and not bound to aliases. *)
(* It is also assumed that c does not contain metas because metas
   *implicitly* depend on Vars but lambda abstraction will not reflect this
   dependency: ?X x = ?1 (?1 is a meta) will return \_.?1 while it should
   return \y. ?1{x\y} (non constant function if ?1 depends on x) (BB) *)
let solve_pattern_eqn env sigma l c =
  let c' = List.fold_right (fun a c ->
    let c' = subst_term sigma (lift 1 (of_alias a)) (lift 1 c) in
    match a with
      (* Rem: if [a] links to a let-in, do as if it were an assumption *)
      | RelAlias n ->
          let open Context.Rel.Declaration in
          let d = map_constr (lift n) (lookup_rel n env) in
          mkLambda_or_LetIn d c'
      | VarAlias id ->
          let d = lookup_named id env in mkNamedLambda_or_LetIn sigma d c'
    )
    l c in
  (* Warning: we may miss some opportunity to eta-reduce more since c'
     is not in normal form *)
  shrink_eta sigma c'

(*****************************************)
(* Refining/solving unification problems *)
(*****************************************)

(* Knowing that [Gamma |- ev : T] and that [ev] is applied to [args],
 * [make_projectable_subst ev args] builds the substitution [Gamma:=args].
 * If a variable and an alias of it are bound to the same instance, we skip
 * the alias (we just use eq_constr -- instead of conv --, since anyway,
 * only instances that are variables -- or evars -- are later considered;
 * moreover, we can bet that similar instances came at some time from
 * the very same substitution. The removal of aliased duplicates is
 * useful to ensure the uniqueness of a projection.
*)

type esubst = {
  ealias : (alias * Id.t) list Int.Map.t;
  evalue : (existential * Id.t) Int.Map.t;
  eindex : Int.Set.t AlsMap.t;
  (** Reverse map of indices in [ealias] containing the corresponding alias *)
}

let make_constructor_subst sigma sign args =
  let rec fold decls args accu = match decls, SList.view args with
  | _ :: _, None | [], Some _ -> assert false
  | [], None -> accu
  | LocalAssum ({ binder_name = id }, _) :: decls, Some (Some a, args) ->
    let accu = fold decls args accu in
    let a', args = decompose_app sigma a in
    begin match EConstr.kind sigma a' with
    | Construct (cstr, _) ->
      let l = try Constrmap.find cstr accu with Not_found -> [] in
      Constrmap.add cstr ((args, id) :: l) accu
    | _ -> accu
    end
  | LocalAssum _ :: decls, Some (None, args) -> fold decls args accu
  | LocalDef _ :: decls, Some (_, args) -> fold decls args accu
  in
  fold sign args Constrmap.empty

let make_projectable_subst aliases sigma sign args =
  let evar_aliases = compute_var_aliases sign sigma in
  (* First compute aliasing equivalence classes *)
  let rec fold accu args decls = match SList.view args, decls with
  | None, _ :: _ | Some _, [] -> assert false
  | None, [] -> accu
  | Some (a, args), decl :: decls ->
    let (i, all, vals, revmap) = fold accu args decls in
    let id = get_id decl in
    let a = match a with None -> mkVar id | Some a -> a in
    let revmap = Id.Map.add id i revmap in
    let oldbindings = match decl with
    | LocalAssum _ -> None
    | LocalDef (_, c, _) ->
      match EConstr.kind sigma c with
      | Var id' ->
        let idc = normalize_alias_var evar_aliases id' in
        let ic, sub = match Id.Map.find_opt idc revmap with
        | Some ic ->
          let bnd = match Int.Map.find_opt ic all with
          | None -> []
          | Some bnd -> bnd
          in
          ic, bnd
        | None ->
          (* [idc] is a filtered variable: treat [id] as an assumption *)
          i, []
        in
        Some (ic, sub)
      | _ -> None
    in
    let all, vals = match oldbindings with
    | None ->
      begin match to_alias sigma a with
      | Some v -> Int.Map.add i [v, id] all, vals
      | None ->
        match destEvar sigma a with
        | ev -> all, Int.Map.add i (ev, id) vals
        | exception DestKO -> all, vals
      end
    | Some (ic, sub) ->
      (* Necessarily a let-binding aliasing a variable *)
      match to_alias sigma a with
      | None -> all, vals
      | Some v ->
        if List.exists (fun (c, _) -> eq_alias v c) sub then all, vals
        else Int.Map.add ic ((v, id) :: sub) all, vals
    in
    (i + 1, all, vals, revmap)
  in
  let (_, ealias, evalue, _) = fold (0, Int.Map.empty, Int.Map.empty, Id.Map.empty) args sign in
  (* Then extract the backpointers. *)
  let fold i bnd eindex =
    let fold accu (a, _) = match AlsMap.find a accu with
    | set -> AlsMap.add a (Int.Set.add i set) accu
    | exception Not_found -> AlsMap.add a (Int.Set.singleton i) accu
    in
    List.fold_left fold eindex bnd
  in
  let eindex = Int.Map.fold fold ealias AlsMap.empty in
  { eindex; ealias; evalue }

(*------------------------------------*
 * operations on the evar constraints *
 *------------------------------------*)

(* We have a unification problem Σ; Γ |- ?e[u1..uq] = t : s where ?e is not yet
 * declared in Σ but yet known to be declarable in some context x1:T1..xq:Tq.
 * [define_evar_from_virtual_equation ... Γ Σ t (x1:T1..xq:Tq) .. (u1..uq) (x1..xq)]
 * declares x1:T1..xq:Tq |- ?e : s such that ?e[u1..uq] = t holds.
 *)

let define_evar_from_virtual_equation define_fun env evd src t_in_env ty_t_in_sign sign filter inst_in_env =
  let (evd, evk) = new_pure_evar sign evd ty_t_in_sign ~filter ~src in
  let t_in_env = whd_evar evd t_in_env in
  let evd = define_fun env evd None (evk, inst_in_env) t_in_env in
  let EvarInfo evi = Evd.find evd evk in
  let inst_in_sign = evar_identity_subst evi in
  let evar_in_sign = mkEvar (evk, inst_in_sign) in
  (evd,whd_evar evd evar_in_sign)

(* We have x1..xq |- ?e1 : τ and had to solve something like
 * Σ; Γ |- ?e1[u1..uq] = (...\y1 ... \yk ... c), where c is typically some
 * ?e2[v1..vn], hence flexible. We had to go through k binders and now
 * virtually have x1..xq, y1'..yk' | ?e1' : τ' and the equation
 * Γ, y1..yk |- ?e1'[u1..uq y1..yk] = c.
 * [materialize_evar Γ evd k (?e1[u1..uq]) τ'] extends Σ with the declaration
 * of ?e1' and returns both its instance ?e1'[x1..xq y1..yk] in an extension
 * of the context of e1 so that e1 can be instantiated by
 * (...\y1' ... \yk' ... ?e1'[x1..xq y1'..yk']),
 * and the instance ?e1'[u1..uq y1..yk] so that the remaining equation
 * ?e1'[u1..uq y1..yk] = c can be registered
 *
 * Note that, because invert_definition does not check types, we need to
 * guess the types of y1'..yn' by inverting the types of y1..yn along the
 * substitution u1..uq.
 *)

exception MorePreciseOccurCheckNeeeded

let materialize_evar define_fun env evd k (evk1,args1) ty_in_env =
  if Evd.is_defined evd evk1 then
      (* Some circularity somewhere (see e.g. #3209) *)
      raise MorePreciseOccurCheckNeeeded;
  let (evk1,args1) = destEvar evd (mkEvar (evk1,args1)) in
  let evi1 = Evd.find_undefined evd evk1 in
  let env1,rel_sign = env_rel_context_chop k env in
  let sign1 = evar_hyps evi1 in
  let filter1 = evar_filter evi1 in
  let src = subterm_source evk1 (Evd.evar_source evi1) in
  let avoid = Environ.ids_of_named_context_val sign1 in
  let inst_in_sign = evar_identity_subst evi1 in
  let open Context.Rel.Declaration in
  let (sign2,filter2,inst2_in_env,inst2_in_sign,_,evd,_) =
    List.fold_right (fun d (sign,filter,inst_in_env,inst_in_sign,env,evd,avoid) ->
      let LocalAssum (na,t_in_env) | LocalDef (na,_,t_in_env) = d in
      let id = map_annot (fun na -> next_name_away na avoid) na in
      let evd,t_in_sign =
        let s = Retyping.get_sort_of env evd t_in_env in
        let evd,ty_t_in_sign = refresh_universes
         ~status:univ_flexible (Some false) env evd (mkSort s) in
        define_evar_from_virtual_equation define_fun env evd src t_in_env
          ty_t_in_sign sign filter inst_in_env in
      let evd,d' = match d with
      | LocalAssum _ -> evd, Context.Named.Declaration.LocalAssum (id,t_in_sign)
      | LocalDef (_,b,_) ->
          let evd,b = define_evar_from_virtual_equation define_fun env evd src b
            t_in_sign sign filter inst_in_env in
          evd, Context.Named.Declaration.LocalDef (id,b,t_in_sign) in
      (push_named_context_val d' sign, Filter.extend 1 filter,
       SList.cons (mkRel 1) (SList.Skip.map (lift 1) inst_in_env),
       SList.cons (mkRel 1) (SList.Skip.map (lift 1) inst_in_sign),
       push_rel d env,evd,Id.Set.add id.binder_name avoid))
      rel_sign
      (sign1,filter1,args1,inst_in_sign,env1,evd,avoid)
  in
  let evd,ev2ty_in_sign =
    let s = Retyping.get_sort_of env evd ty_in_env in
    let evd,ty_t_in_sign = refresh_universes
     ~status:univ_flexible (Some false) env evd (mkSort s) in
    define_evar_from_virtual_equation define_fun env evd src ty_in_env
      ty_t_in_sign sign2 filter2 inst2_in_env in
  let (evd, ev2_in_sign) =
    new_pure_evar sign2 evd ev2ty_in_sign ~filter:filter2 ~src in
  let ev2_in_env = (ev2_in_sign, inst2_in_env) in
  (evd, mkEvar (ev2_in_sign, inst2_in_sign), ev2_in_env)

let restrict_upon_filter evd evk p args =
  let oldfullfilter = evar_filter (Evd.find_undefined evd evk) in
  let args = Array.of_list args in
  let len = Array.length args in
  Filter.restrict_upon oldfullfilter len (fun i -> p (Array.unsafe_get args i))

let check_evar_instance_evi unify flags env evd evi body =
  let evenv = evar_env env evi in
  (* FIXME: The body might be ill-typed when this is called from w_merge *)
  (* This happens in practice, cf MathClasses build failure on 2013-3-15 *)
  let ty =
    try Retyping.get_type_of ~lax:true evenv evd body
    with Retyping.RetypeError _ ->
      let loc, _ = Evd.evar_source evi in user_err ?loc (Pp.(str "Ill-typed evar instance"))
  in
  match unify flags TypeUnification evenv evd Conversion.CUMUL ty (Evd.evar_concl evi) with
  | Success evd -> evd
  | UnifFailure _ -> raise (IllTypedInstance (evenv,evd,ty, Evd.evar_concl evi))

let check_evar_instance unify flags env evd evk body =
  let evi = try Evd.find_undefined evd evk with Not_found -> assert false in
  check_evar_instance_evi unify flags env evd evi body

(***************)
(* Unification *)

(* Inverting constructors in instances (common when inferring type of match) *)

let find_projectable_constructor env evd cstr k args cstr_subst =
  try
    let l = Constrmap.find cstr cstr_subst in
    let args = Array.map (lift (-k)) args in
    let l =
      List.filter (fun (args',id) ->
        (* is_conv is maybe too strong (and source of useless computation) *)
        (* (at least expansion of aliases is needed) *)
        Array.for_all2 (fun c1 c2 -> is_conv env evd c1 c2) args args') l in
    List.map snd l
  with Not_found ->
    []

(* [find_projectable_vars env sigma y subst] finds all vars of [subst]
 * that project on [y]. It is able to find solutions to the following
 * two kinds of problems:
 *
 * - ?n[...;x:=y;...] = y
 * - ?n[...;x:=?m[args];...] = y with ?m[args] = y recursively solvable
 *
 * (see test-suite/success/Fixpoint.v for an example of application of
 * the second kind of problem).
 *
 * The seek for [y] is up to variable aliasing.  In case of solutions that
 * differ only up to aliasing, the binding that requires the less
 * steps of alias reduction is kept. At the end, only one solution up
 * to aliasing is kept.
 *
 * [find_projectable_vars] also unifies against evars that themselves mention
 * [y] and recursively.
 *
 * In short, the following situations give the following solutions:
 *
 * problem                        evar ctxt   soluce remark
 * z1; z2:=z1 |- ?ev[z1;z2] = z1  y1:A; y2:=y1  y1  \ thanks to defs kept in
 * z1; z2:=z1 |- ?ev[z1;z2] = z2  y1:A; y2:=y1  y2  / subst and preferring =
 * z1; z2:=z1 |- ?ev[z1]    = z2  y1:A          y1  thanks to expand_var
 * z1; z2:=z1 |- ?ev[z2]    = z1  y1:A          y1  thanks to expand_var
 * z3         |- ?ev[z3;z3] = z3  y1:A; y2:=y1  y2  see make_projectable_subst
 *
 * Remark: [find_projectable_vars] assumes that identical instances of
 * variables in the same set of aliased variables are already removed (see
 * [make_projectable_subst])
 *)

type evar_projection =
| ProjectVar
| ProjectEvar of EConstr.existential * undefined evar_info * Id.t * evar_projection

exception NotUnique
exception NotUniqueInType of (Id.t * evar_projection) list

let rec assoc_up_to_alias sigma aliases y = function
  | [] -> assert false
  | (c, id)::l ->
    if eq_alias c y then id
    else assoc_up_to_alias sigma aliases y l

let rec find_projectable_vars aliases sigma y subst =
  let indices = try AlsMap.find y subst.eindex with Not_found -> Int.Set.empty in
  let is_projectable_var i subst1 =
    (* First test if some [id] aliased to [idc] is bound to [y] in [subst] *)
    let idcl = Int.Map.find i subst.ealias in
    let id = assoc_up_to_alias sigma aliases y idcl in
    (id, ProjectVar)::subst1
  in
  let is_projectable_evar i (c, id) subst2 =
    (* Then test if [idc] is (indirectly) bound in [subst] to some evar *)
    (* projectable on [y] *)
    if Int.Set.mem i indices then subst2 (* already found by is_projectable_var *)
    else if Evd.is_defined sigma (fst c) then subst2 (* already solved *)
    else
      let (evk,argsv as t) = c in
      let evi = Evd.find_undefined sigma evk in
      let subst = make_projectable_subst aliases sigma (evar_filtered_context evi) argsv in
      let l = find_projectable_vars aliases sigma y subst in
      match l with
      | [id',p] -> (id, ProjectEvar (t, evi, id', p)) :: subst2
      | _ -> subst2
  in
  let subst1 = Int.Set.fold is_projectable_var indices [] in
  let subst2 = Int.Map.fold is_projectable_evar subst.evalue [] in
  (* We return the substitution with ProjectVar first (from most
     recent to oldest var), followed by ProjectEvar (from most recent
     to oldest var too) *)
  subst1 @ subst2

(* [filter_solution] checks if one and only one possible projection exists
 * among a set of solutions to a projection problem *)

let filter_solution = function
  | [] -> raise Not_found
  | _ :: _ :: _ -> raise NotUnique
  | [id] -> mkVar id

let project_with_effects aliases sigma t subst =
  let indices = AlsMap.find t subst.eindex in
  let is_projectable i accu =
    let idcl = Int.Map.find i subst.ealias in
    assoc_up_to_alias sigma aliases t idcl :: accu
  in
  filter_solution (Int.Set.fold is_projectable indices [])

(* In case the solution to a projection problem requires the instantiation of
 * subsidiary evars, [do_projection_effects] performs them; it
 * also try to instantiate the type of those subsidiary evars if their
 * type is an evar too.
 *
 * Note: typing creates new evar problems, which induces a recursive dependency
 * with [define]. To avoid a too large set of recursive functions, we
 * pass [define] to [do_projection_effects] as a parameter.
 *)

let rec do_projection_effects unify flags define_fun env ty evd = function
  | ProjectVar -> evd
  | ProjectEvar ((evk,argsv),evi,id,p) ->
      let evd = check_evar_instance unify flags env evd evk (mkVar id) in
      let evd = Evd.define evk (EConstr.mkVar id) evd in
      (* TODO: simplify constraints involving evk *)
      let evd = do_projection_effects unify flags define_fun env ty evd p in
      let ty = whd_all env evd (Lazy.force ty) in
      if not (isSort evd ty) then
        (* Don't try to instantiate if a sort because if evar_concl is an
           evar it may commit to a univ level which is not the right
           one (however, regarding coercions, because t is obtained by
           unif, we know that no coercion can be inserted) *)
        let ty' = instantiate_evar_array evd evi (Evd.evar_concl evi) argsv in
        if isEvar evd ty' then define_fun env evd (Some false) (destEvar evd ty') ty else evd
      else
        evd

(* Assuming Σ; Γ, y1..yk |- c, [invert_arg_from_subst Γ k Σ [x1:=u1..xn:=un] c]
 * tries to return φ(x1..xn) such that equation φ(u1..un) = c is valid.
 * The strategy is to imitate the structure of c and then to invert
 * the variables of c (i.e. rels or vars of Γ) using the algorithm
 * implemented by project_with_effects/find_projectable_vars.
 * It returns either a unique solution or says whether 0 or more than
 * 1 solutions is found.
 *
 * Precondition:  Σ; Γ, y1..yk |- c /\ Σ; Γ |- u1..un
 * Postcondition: if φ(x1..xn) is returned then
 *                Σ; Γ, y1..yk |- φ(u1..un) = c /\ x1..xn |- φ(x1..xn)
 *
 * The effects correspond to evars instantiated while trying to project.
 *
 * [invert_arg_from_subst] is used on instances of evars. Since the
 * evars are flexible, these instances are potentially erasable. This
 * is why we don't investigate whether evars in the instances of evars
 * are unifiable, to the contrary of [invert_definition].
 *)

type projectibility_kind =
  | NoUniqueProjection
  | UniqueProjection of EConstr.constr

type projectibility_status =
  | CannotInvert
  | Invertible of projectibility_kind

let invert_arg_from_subst evd aliases k0 subst_in_env_extended_with_k_binders c_in_env_extended_with_k_binders =
  let rec aux k t =
    match EConstr.kind evd t with
    | Rel i when i>k0+k -> aux' k (RelAlias (i-k))
    | Var id -> aux' k (VarAlias id)
    | _ -> map_with_binders evd succ aux k t
  and aux' k t =
    try project_with_effects aliases evd t subst_in_env_extended_with_k_binders
    with Not_found ->
      match expand_alias_once aliases t with
      | None -> raise Not_found
      | Some c -> aux k (Alias.eval (Alias.lift k c)) in
  try
    let c = aux 0 c_in_env_extended_with_k_binders in
    Invertible (UniqueProjection c)
  with
    | Not_found -> CannotInvert
    | NotUnique -> Invertible NoUniqueProjection

let invert_arg fullenv evd aliases k evk subst_in_env_extended_with_k_binders c_in_env_extended_with_k_binders =
  let res = invert_arg_from_subst evd aliases k subst_in_env_extended_with_k_binders c_in_env_extended_with_k_binders in
  match res with
  | Invertible (UniqueProjection c) when not (noccur_evar fullenv evd evk c)
      ->
      CannotInvert
  | _ ->
      res

exception NotEnoughInformationToInvert

let extract_unique_projection = function
| Invertible (UniqueProjection c) -> c
| _ ->
    (* For instance, there are evars with non-invertible arguments and *)
    (* we cannot arbitrarily restrict these evars before knowing if there *)
    (* will really be used; it can also be due to some argument *)
    (* (typically a rel) that is not inversible and that cannot be *)
    (* inverted either because it is needed for typing the conclusion *)
    (* of the evar to project *)
  raise NotEnoughInformationToInvert

let extract_candidates sols =
  try
    UpdateWith
      (List.map (function (id,ProjectVar) -> mkVar id | _ -> raise_notrace Exit) sols)
  with Exit ->
    NoUpdate

let invert_invertible_arg fullenv evd aliases k (evk,argsv) args' =
  let evi = Evd.find_undefined evd evk in
  let subst = make_projectable_subst aliases evd (evar_filtered_context evi) argsv in
  let invert arg =
    let p = invert_arg fullenv evd aliases k evk subst arg in
    extract_unique_projection p
  in
  List.map invert args'

(* Redefines an evar with a smaller context (i.e. it may depend on less
 * variables) such that c becomes closed.
 * Example: in "fun (x:?1) (y:list ?2[x]) => x = y :> ?3[x,y] /\ x = nil bool"
 * ?3 <-- ?1          no pb: env of ?3 is larger than ?1's
 * ?1 <-- list ?2     pb: ?2 may depend on x, but not ?1.
 * What we do is that ?2 is defined by a new evar ?4 whose context will be
 * a prefix of ?2's env, included in ?1's env.
 *
 * If "hyps |- ?e : T" and "filter" selects a subset hyps' of hyps then
 * [do_restrict_hyps evd ?e filter] sets ?e:=?e'[hyps'] and returns ?e'
 * such that "hyps' |- ?e : T"
 *)

let set_of_evctx l =
  List.fold_left (fun s decl -> Id.Set.add (get_id decl) s) Id.Set.empty l

let filter_effective_candidates evd evi filter candidates =
  match filter with
  | None -> candidates
  | Some filter ->
      let ids = set_of_evctx (Filter.filter_list filter (evar_context evi)) in
      List.filter (fun a -> Id.Set.subset (collect_vars evd a) ids) candidates

let filter_candidates evd evk filter candidates_update =
  let evi = Evd.find_undefined evd evk in
  let candidates = match candidates_update with
  | NoUpdate -> Evd.evar_candidates evi
  | UpdateWith c -> Some c
  in
  match candidates with
  | None -> NoUpdate
  | Some l ->
      let l' = filter_effective_candidates evd evi filter l in
      if List.length l = List.length l' && candidates_update = NoUpdate then
        NoUpdate
      else
        UpdateWith l'

(* Given a filter refinement for the evar [evk], restrict it so that
   dependencies are preserved *)

let closure_of_filter ~can_drop evd evk = function
  | None -> None
  | Some filter ->
  let evi = Evd.find_undefined evd evk in
  let vars = collect_vars evd (evar_concl evi) in
  let test b decl = b || Id.Set.mem (get_id decl) vars ||
                    match decl with
                    | LocalAssum _ ->
                       false
                    | LocalDef (_,c,_) ->
                       not (can_drop || isRel evd c || isVar evd c)
  in
  let newfilter = Filter.map_along test filter (evar_context evi) in
  (* Now ensure that restriction is at least what is was originally *)
  let newfilter = Option.cata (Filter.map_along (&&) newfilter) newfilter (Filter.repr (evar_filter evi)) in
  if Filter.equal newfilter (evar_filter evi) then None else Some newfilter

(* The filter is assumed to be at least stronger than the original one *)
let restrict_hyps ~can_drop evd evk filter candidates =
    (* What to do with dependencies?
       Assume we have x:A, y:B(x), z:C(x,y) |- ?e:T(x,y,z) and restrict on y.
       - If y is in a non-erasable position in C(x,y) (i.e. it is not below an
         occurrence of x in the hnf of C), then z should be removed too.
       - If y is in a non-erasable position in T(x,y,z) then the problem is
         unsolvable.
       Computing whether y is erasable or not may be costly and the
       interest for this early detection in practice is not obvious. We let
       it for future work. In any case, thanks to the use of filters, the whole
       (unrestricted) context remains consistent. *)
    let candidates = filter_candidates evd evk (Some filter) candidates in
    let typablefilter = closure_of_filter ~can_drop evd evk (Some filter) in
    (typablefilter,candidates)

exception EvarSolvedWhileRestricting of evar_map * EConstr.constr

let do_restrict_hyps ~can_drop evd (evk,args as ev) filter candidates =
  let filter,candidates = match filter with
  | None -> None,candidates
  | Some filter -> restrict_hyps ~can_drop evd evk filter candidates in
  match candidates,filter with
  | UpdateWith [], _ -> user_err Pp.(str "Not solvable.")
  | UpdateWith [nc],_ ->
      let evd = Evd.define evk nc evd in
      raise (EvarSolvedWhileRestricting (evd,mkEvar ev))
  | NoUpdate, None -> evd,ev
  | _ -> restrict_applied_evar evd ev filter candidates

(* [postpone_non_unique_projection] postpones equation of the form ?e[?] = c *)
(* ?e is assumed to have no candidates *)

let postpone_non_unique_projection env evd pbty (evk,argsv as ev) sols rhs =
  let rhs = expand_vars_in_term env evd rhs in
  let filter a = match EConstr.kind evd a with
  | Rel n -> not (noccurn evd  n rhs)
  | Var id ->
    local_occur_var evd id rhs
      || List.exists (fun (id', _) -> Id.equal id id') sols
  | _ -> true
  in
  let argsv = Evd.expand_existential evd (evk, argsv) in
  let filter = restrict_upon_filter evd evk filter argsv in
      (* Keep only variables that occur in rhs *)
      (* This is not safe: is the variable is a local def, its body *)
      (* may contain references to variables that are removed, leading to *)
      (* an ill-formed context. We would actually need a notion of filter *)
      (* that says that the body is hidden. Note that expand_vars_in_term *)
      (* expands only rels and vars aliases, not rels or vars bound to an *)
      (* arbitrary complex term *)
  let filter = closure_of_filter ~can_drop:false evd evk filter in
  let candidates = extract_candidates sols in
  match candidates with
  | NoUpdate ->
    (* We made an approximation by not expanding a local definition *)
    let evd,ev = restrict_applied_evar evd ev filter NoUpdate in
    let pb = (pbty,env,mkEvar ev,rhs) in
    add_conv_oriented_pb pb evd
  | UpdateWith c ->
    restrict_evar evd evk filter (UpdateWith c)

(* [solve_evar_evar f Γ Σ ?e1[u1..un] ?e2[v1..vp]] applies an heuristic
 * to solve the equation Σ; Γ ⊢ ?e1[u1..un] = ?e2[v1..vp]:
 * - if there are at most one φj for each vj s.t. vj = φj(u1..un),
 *   we first restrict ?e2 to the subset v_k1..v_kq of the vj that are
 *   inversible and we set ?e1[x1..xn] := ?e2[φk1(x1..xn)..φkp(x1..xn)]
 *   (this is a case of pattern-unification)
 * - symmetrically if there are at most one ψj for each uj s.t.
 *   uj = ψj(v1..vp),
 * - otherwise, each position i s.t. ui does not occur in v1..vp has to
 *   be restricted and similarly for the vi, and we leave the equation
 *   as an open equation (performed by [postpone_evar])
 *
 * Warning: the notion of unique φj is relative to some given class
 * of unification problems
 *
 * Note: argument f is the function used to instantiate evars.
 *)

let filter_compatible_candidates unify flags env evd evi args rhs c =
  let c' = instantiate_evar_array evd evi c args in
  match unify flags TermUnification env evd Conversion.CONV rhs c' with
  | Success evd -> Inl (c,evd)
  | UnifFailure _ -> Inr c'

(* [restrict_candidates ... filter ev1 ev2] restricts the candidates
   of ev1, removing those not compatible with the filter, as well as
   those not convertible to some candidate of ev2 *)

exception DoesNotPreserveCandidateRestriction

let restrict_candidates unify flags env evd filter1 (evk1,argsv1) (evk2,argsv2) =
  let evi1 = Evd.find_undefined evd evk1 in
  let evi2 = Evd.find_undefined evd evk2 in
  match Evd.evar_candidates evi1, Evd.evar_candidates evi2 with
  | _, None -> filter_candidates evd evk1 filter1 NoUpdate
  | None, Some _ -> raise DoesNotPreserveCandidateRestriction
  | Some l1, Some l2 ->
      let l1 = filter_effective_candidates evd evi1 filter1 l1 in
      let l1' = List.filter (fun c1 ->
        let c1' = instantiate_evar_array evd evi1 c1 argsv1 in
        let filter c2 =
          let compatibility = filter_compatible_candidates unify flags env evd evi2 argsv2 c1' c2 in
          match compatibility with
          | Inl _ -> true
          | Inr _ -> false
        in
        let filtered = List.filter filter l2 in
        match filtered with [] -> false | _ -> true) l1 in
      if Int.equal (List.length l1) (List.length l1') then NoUpdate
      else UpdateWith l1'

exception CannotProject of evar_map * EConstr.existential

(* Assume that FV(?n[x1:=t1..xn:=tn]) belongs to some set U.
   Can ?n be instantiated by a term u depending essentially on xi such that the
   FV(u[x1:=t1..xn:=tn]) are in the set U?
   - If ti is a variable, it has to be in U.
   - If ti is a constructor, its parameters cannot be erased even if u
     matches on it, so we have to discard ti if the parameters
     contain variables not in U.
   - If ti is rigid, we have to discard it if it contains variables in U.

  Note: when restricting as part of an equation ?n[x1:=t1..xn:=tn] = ?m[...]
  then, occurrences of ?m in the ti can be seen, like variables, as occurrences
  of subterms to eventually discard so as to be allowed to keep ti.
*)

let rec is_constrainable_in top env evd k (evk,(fv_rels,fv_ids) as g) t =
  let f,args = decompose_app evd t in
  match EConstr.kind evd f with
  | Construct ((ind,_),u) ->
    let n = Inductiveops.inductive_nparams env ind in
    if n > Array.length args then true (* We don't try to be more clever *)
    else
      let params = fst (Array.chop n args) in
      Array.for_all (is_constrainable_in false env evd k g) params
  | Ind _ -> Array.for_all (is_constrainable_in false env evd k g) args
  | Prod (na,t1,t2) -> is_constrainable_in false env evd k g t1 && is_constrainable_in false env evd k g t2
  | Evar (evk',_ as ev') ->
    (*If ev' needed, one may also try to restrict it*)
    top || not (Evar.equal evk' evk || occur_evar evd evk (Evd.existential_type evd ev'))
  | Var id -> Id.Set.mem id fv_ids
  | Rel n -> n <= k || Int.Set.mem n fv_rels
  | Sort _ -> true
  | _ -> (* We don't try to be more clever *) true

let has_constrainable_free_vars env evd aliases force k ev (fv_rels,fv_ids,let_rels,let_ids) t =
  match to_alias evd t with
  | Some t ->
    let expanded, _, _ = expansion_of_var evd aliases t in
    if expanded then
    (* t is a local definition, we keep it only if appears in the list *)
    (* of let-in variables effectively occurring on the right-hand side, *)
    (* which is the only reason to keep it when inverting arguments *)
      match t with
      | VarAlias id -> Id.Set.mem id let_ids
      | RelAlias n -> Int.Set.mem n let_rels
    else begin match t with
    | VarAlias id -> Id.Set.mem id fv_ids
    | RelAlias n -> n <= k || Int.Set.mem n fv_rels
    end
  | None ->
    (* t is an instance for a proper variable; we filter it along *)
    (* the free variables allowed to occur *)
    (not force || noccur_evar env evd ev t) && is_constrainable_in true env evd k (ev,(fv_rels,fv_ids)) t

exception EvarSolvedOnTheFly of evar_map * EConstr.constr

(* Try to project evk1[argsv1] on evk2[argsv2], if [ev1] is a pattern on
   the common domain of definition *)
let project_evar_on_evar force unify flags env evd aliases k2 pbty (evk1,argsv1 as ev1) (evk2,argsv2 as ev2) =
  if Option.is_empty pbty && SList.is_default argsv2 &&
    (* This ensures that the named context of [evk2] is a permutation of the one
       from [env]. In particular its filter must be trivial. *)
    Int.equal (SList.length argsv2) (Range.length (Environ.named_context_val env).env_named_idx) &&
    SList.Skip.for_all (fun arg -> noccur_evar env evd evk2 arg && closed0 evd arg) argsv1 &&
    let evi2 = Evd.find_undefined evd evk2 in Option.is_empty (Evd.evar_candidates evi2)
  then
    evd, EConstr.mkEvar ev1
  else
  (* Apply filtering on ev1 so that fvs(ev1) are in fvs(ev2). *)
  let fvs2 = free_vars_and_rels_up_alias_expansion env evd aliases (mkEvar ev2) in
  let argsv1 = Evd.expand_existential evd ev1 in
  let filter1 = restrict_upon_filter evd evk1
    (has_constrainable_free_vars env evd aliases force k2 evk2 fvs2)
    argsv1 in
  let candidates1 =
    try restrict_candidates unify flags env evd filter1 ev1 ev2
    with DoesNotPreserveCandidateRestriction ->
      let evd,ev1' = do_restrict_hyps ~can_drop:force evd ev1 filter1 NoUpdate in
      raise (CannotProject (evd,ev1')) in
  let evd,(evk1',args1 as ev1') =
    try do_restrict_hyps ~can_drop:force evd ev1 filter1 candidates1
    with EvarSolvedWhileRestricting (evd,ev1) ->
      raise (EvarSolvedOnTheFly (evd,ev1)) in
  (* Only try pruning on variable substitutions, postpone otherwise. *)
  (* Rules out non-linear instances. *)
  if Option.is_empty pbty && is_unification_pattern_pure_evar env evd ev2 (mkEvar ev1) then
    try
      let args1 = Evd.expand_existential evd ev1' in
      evd, EConstr.mkLEvar evd (evk1', invert_invertible_arg env evd aliases k2 ev2 args1)
    with NotEnoughInformationToInvert ->
      raise (CannotProject (evd,ev1'))
  else
    raise (CannotProject (evd,ev1'))

let update_evar_info ev1 ev2 evd =
  (* We update the source of obligation evars during evar-evar unifications. *)
  let EvarInfo evi1 = Evd.find evd ev1 in
  let loc, evs1 = evar_source evi1 in
  Evd.update_source evd ev2 (loc, evs1)

let solve_evar_evar_l2r force f unify flags env evd aliases pbty ev1 (evk2,_ as ev2) =
  try
    let evd,body = project_evar_on_evar force unify flags env evd aliases 0 pbty ev1 ev2 in
    let evd =
      if is_obligation_evar evd evk2 then
        update_evar_info evk2 (fst (destEvar evd body)) evd
      else evd
    in
    let evi = Evd.find_undefined evd evk2 in
    let evd' = Evd.define_with_evar evk2 body evd in
    check_evar_instance_evi unify flags env evd' evi body
  with EvarSolvedOnTheFly (evd,c) ->
    f env evd pbty ev2 c

let opp_problem = function None -> None | Some b -> Some (not b)

let preferred_orientation evd evk1 evk2 =
  if is_obligation_evar evd evk1 then true
  else if is_obligation_evar evd evk2 then false
  else true

let solve_evar_evar_aux force f unify flags env evd pbty (evk1,args1 as ev1) (evk2,args2 as ev2) =
  let aliases = make_alias_map env evd in
  let allowed_ev1 = is_evar_allowed flags evk1 in
  let allowed_ev2 = is_evar_allowed flags evk2 in
  if preferred_orientation evd evk1 evk2 then
    try if allowed_ev1 then
        solve_evar_evar_l2r force f unify flags env evd aliases (opp_problem pbty) ev2 ev1
      else raise (CannotProject (evd,ev2))
    with CannotProject (evd,ev2) ->
      try if allowed_ev2 then
          solve_evar_evar_l2r force f unify flags env evd aliases pbty ev1 ev2
        else raise (CannotProject (evd,ev1))
      with CannotProject (evd,ev1) ->
        add_conv_oriented_pb ~tail:true (pbty,env,mkEvar ev1,mkEvar ev2) evd
  else
    try if allowed_ev2 then
        solve_evar_evar_l2r force f unify flags env evd aliases pbty ev1 ev2
      else raise (CannotProject (evd,ev1))
    with CannotProject (evd,ev1) ->
    try if allowed_ev1 then
        solve_evar_evar_l2r force f unify flags env evd aliases (opp_problem pbty) ev2 ev1
      else raise (CannotProject (evd,ev2))
    with CannotProject (evd,ev2) ->
      add_conv_oriented_pb ~tail:true (pbty,env,mkEvar ev1,mkEvar ev2) evd

(** Precondition: evk1 is not frozen *)
let solve_evar_evar ?(force=false) f unify flags env evd pbty (evk1,args1 as ev1) (evk2,args2 as ev2) =
  let pbty = if force then None else pbty in
  let evi = Evd.find_undefined evd evk1 in
  let downcast evk t evd = downcast evk t evd in
  let evd =
    try
      (* ?X : Π Δ. Type i = ?Y : Π Δ'. Type j.
         The body of ?X and ?Y just has to be of type Π Δ. Type k for some k <= i, j. *)
      let evienv = Evd.evar_env env evi in
      let ctx1, i = Reductionops.dest_arity evienv evd (Evd.evar_concl evi) in
      let evi2 = Evd.find_undefined evd evk2 in
      let evi2env = Evd.evar_env env evi2 in
      let ctx2, j = Reductionops.dest_arity evi2env evd (Evd.evar_concl evi2) in
        if i == j || Evd.check_eq evd i j
        then (* Shortcut, i = j *)
          evd
        else if Evd.check_leq evd i j then
          let t2 = it_mkProd_or_LetIn (mkSort i) ctx2 in
          downcast evk2 t2 evd
        else if Evd.check_leq evd j i then
          let t1 = it_mkProd_or_LetIn (mkSort j) ctx1 in
          downcast evk1 t1 evd
        else
          let evd, k = Evd.new_sort_variable univ_flexible_alg evd in
          let t1 = it_mkProd_or_LetIn (mkSort k) ctx1 in
          let t2 = it_mkProd_or_LetIn (mkSort k) ctx2 in
          let evd = Evd.set_leq_sort env (Evd.set_leq_sort env evd k i) k j in
          downcast evk2 t2 (downcast evk1 t1 evd)
    with Reduction.NotArity ->
      evd in
  solve_evar_evar_aux force f unify flags env evd pbty ev1 ev2

(* Solve pbs ?e[t1..tn] = ?e[u1..un] which arise often in fixpoint
 * definitions. We try to unify the ti with the ui pairwise. The pairs
 * that don't unify are discarded (i.e. ?e is redefined so that it does not
 * depend on these args). *)

let solve_refl ?(can_drop=false) unify flags env evd pbty evk argsv1 argsv2 =
  let evdref = ref evd in
  let eq_constr c1 c2 = match EConstr.eq_constr_universes env !evdref c1 c2 with
  | None -> false
  | Some cstr ->
    try evdref := Evd.add_universe_constraints !evdref cstr; true
    with UniversesDiffer -> false
  in
  let argsv1e = Evd.expand_existential !evdref (evk, argsv1) in
  let argsv2e = Evd.expand_existential !evdref (evk, argsv2) in
  if List.equal eq_constr argsv1e argsv2e then !evdref else
  (* Filter and restrict if needed *)
  let args = List.map2 (fun a1 a2 -> (a1, a2)) argsv1e argsv2e in
  let untypedfilter =
    restrict_upon_filter evd evk
      (fun (a1,a2) -> unify flags TermUnification env evd Conversion.CONV a1 a2) args in
  let candidates = filter_candidates evd evk untypedfilter NoUpdate in
  let filter = closure_of_filter ~can_drop:false evd evk untypedfilter in
  let evd',ev1 = restrict_applied_evar evd (evk, argsv1) filter candidates in
  let allowed = is_evar_allowed flags evk in
  if Evar.equal (fst ev1) evk && (not allowed || can_drop) then
    (* No refinement needed *) evd'
  else
    (* either progress, or not allowed to drop, e.g. to preserve possibly *)
    (* informative equations such as ?e[x:=?y]=?e[x:=?y'] where we don't know *)
    (* if e can depend on x until ?y is not resolved, or, conversely, we *)
    (* don't know if ?y has to be unified with ?y, until e is resolved *)
  if not allowed then
    (* We cannot prune a frozen evar *)
    add_conv_oriented_pb (pbty,env,mkEvar (evk, argsv1),mkEvar (evk, argsv2)) evd
  else
    let argsv2 = restrict_instance evd' evk filter argsv2 in
    let ev2 = (fst ev1,argsv2) in
    (* Leave a unification problem *)
    add_conv_oriented_pb (pbty,env,mkEvar ev1,mkEvar ev2) evd'

(* If the evar can be instantiated by a finite set of candidates known
   in advance, we check which of them apply *)

exception NoCandidates
exception IncompatibleCandidates of EConstr.t

let solve_candidates unify flags env evd (evk,argsv) rhs =
  let evi = Evd.find_undefined evd evk in
  match Evd.evar_candidates evi with
  | None -> raise NoCandidates
  | Some l ->
      let rec aux = function
        | [] -> [], []
        | c::l ->
           let compatl, disjointl = aux l in
           match filter_compatible_candidates unify flags env evd evi argsv rhs c with
           | Inl c -> c::compatl, disjointl
           | Inr c -> compatl, c::disjointl in
      match aux l with
      | [], c::_ -> raise (IncompatibleCandidates c)
      | [c,evd], _ ->
          (* solve_candidates might have been called recursively in the mean *)
          (* time and the evar been solved by the filtering process *)
         if Evd.is_undefined evd evk then
           let evd = check_evar_instance unify flags env evd evk c in
           let evd = Evd.define evk c evd in
           evd
         else evd
      | l, _::_  (* At least one discarded candidate *) ->
          let candidates = List.map fst l in
          restrict_evar evd evk None (UpdateWith candidates)
      | l, [] -> evd

let occur_evar_upto_types sigma n c =
  let seen = ref Evar.Set.empty in
  let rec occur_rec c = match EConstr.kind sigma c with
    | Evar (sp,_) when Evar.equal sp n -> raise Occur
    | Evar (sp,args as e) ->
       if Evar.Set.mem sp !seen then
         SList.Skip.iter occur_rec args
       else (
         seen := Evar.Set.add sp !seen;
         occur_rec (Evd.existential_type sigma e))
    | _ -> EConstr.iter sigma occur_rec c
  in
  try occur_rec c; false with Occur -> true

let instantiate_evar unify flags env evd evk body =
  (* Check instance freezing the evar to be defined, as
     checking could involve the same evar definition problem again otherwise *)
  let allowed_evars = AllowedEvars.remove evk flags.allowed_evars in
  let flags = { flags with allowed_evars } in
  let evd' = check_evar_instance unify flags env evd evk body in
  Evd.define evk body evd'

(* We try to instantiate the evar assuming the body won't depend
 * on arguments that are not Rels or Vars, or appearing several times
 * (i.e. we tackle a generalization of Miller-Pfenning patterns unification)
 *
 * 1) Let "env |- ?ev[hyps:=args] = rhs" be the unification problem
 * 2) We limit it to a patterns unification problem "env |- ev[subst] = rhs"
 *    where only Rel's and Var's are relevant in subst
 * 3) We recur on rhs, "imitating" the term, and failing if some Rel/Var is
 *    not in the scope of ?ev. For instance, the problem
 *    "y:nat |- ?x[] = y" where "|- ?1:nat" is not satisfiable because
 *    ?1 would be instantiated by y which is not in the scope of ?1.
 * 4) We try to "project" the term if the process of imitation fails
 *    and that only one projection is possible
 *
 * Note: we don't assume rhs in normal form, it may fail while it would
 * have succeeded after some reductions.
 *
 * This is the work of [invert_definition Γ Σ ?ev[hyps:=args] c]
 * Precondition:  Σ; Γ, y1..yk |- c /\ Σ; Γ |- u1..un
 * Postcondition: if φ(x1..xn) is returned then
 *                Σ; Γ, y1..yk |- φ(u1..un) = c /\ x1..xn |- φ(x1..xn)
 *)

exception NotInvertibleUsingOurAlgorithm of EConstr.constr
exception NotEnoughInformationToProgress of (Id.t * evar_projection) list
exception NotEnoughInformationEvarEvar of EConstr.constr
exception OccurCheckIn of evar_map * EConstr.constr
exception MetaOccurInBodyInternal

let rec invert_definition unify flags choose imitate_defs
                          env evd pbty (evk,argsv as ev) rhs =
  let EvarInfo evi = Evd.find evd evk in
  let sign = evar_filtered_context evi in
  let rhs = whd_beta env evd rhs (* heuristic *) in
  let fast =
    let names = ref Id.Set.empty in
    let rec is_id_subst ctxt s =
      match ctxt, SList.view s with
      | (decl :: ctxt'), Some (c, s') ->
        let id = get_id decl in
        names := Id.Set.add id !names;
        (match c with None -> true | Some c -> isVarId evd id c) && is_id_subst ctxt' s'
      | [], None -> true
      | _ -> false
    in
      is_id_subst sign argsv &&
      closed0 evd rhs &&
      Id.Set.subset (collect_vars evd rhs) !names
  in

  if fast then (evd, nf_evar evd rhs) (* FIXME? *)
  else

  let evdref = ref evd in
  let progress = ref false in
  let aliases = make_alias_map env evd in
  let subst = make_projectable_subst aliases evd sign argsv in
  let cstr_subst = make_constructor_subst evd sign argsv in

  (* Projection *)
  let project_variable t =
    (* Evar/Var problem: unifiable iff variable projectable from ev subst *)
    try
      let sols = find_projectable_vars aliases !evdref t subst in
      let c, p = match sols with
        | [] -> raise Not_found
        | [id,p] -> (mkVar id, p)
        | (id,p)::_ ->
            if choose then (mkVar id, p) else raise (NotUniqueInType sols)
      in
      let ty = lazy (Retyping.get_type_of env !evdref (of_alias t)) in
      let evd = do_projection_effects unify flags (evar_define unify flags ~choose) env ty !evdref p in
      evdref := evd;
      c
    with
      | Not_found -> raise (NotInvertibleUsingOurAlgorithm (of_alias t))
      | NotUniqueInType sols ->
          if not !progress then
            raise (NotEnoughInformationToProgress sols);
          (* No unique projection but still restrict to where it is possible *)
          (* materializing is necessary, but is restricting useful? *)
          let t' = of_alias t in
          let ty = Retyping.get_type_of env !evdref t' in
          let (evd,evar,(evk',argsv' as ev')) =
            materialize_evar (evar_define unify flags ~choose) env !evdref 0 ev ty in
          let ts = expansions_of_var aliases t in
          let test c = isEvar evd c || List.exists (is_alias evd c) ts in
          let argsv' = Evd.expand_existential evd (evk, argsv') in
          let filter = restrict_upon_filter evd evk test argsv' in
          let filter = closure_of_filter ~can_drop:choose evd evk' filter in
          let candidates = extract_candidates sols in
          let evd = match candidates with
          | NoUpdate ->
            let evd, ev'' = restrict_applied_evar evd ev' filter NoUpdate in
            add_conv_oriented_pb ~tail:false (None,env,mkEvar ev'',t') evd
          | UpdateWith _ ->
            restrict_evar evd evk' filter candidates
          in
          evdref := evd;
          evar in

  let rec imitate (env',k as envk) t =
    match EConstr.kind !evdref t with
    | Rel i when i>k ->
        let open Context.Rel.Declaration in
        (match Environ.lookup_rel i env' with
        | LocalAssum _ -> project_variable (RelAlias (i-k))
        | LocalDef (_,b,_) ->
          try project_variable (RelAlias (i-k))
          with NotInvertibleUsingOurAlgorithm _ when imitate_defs ->
            imitate envk (lift i (EConstr.of_constr b)))
    | Var id ->
        (match Environ.lookup_named id env' with
        | LocalAssum _ -> project_variable (VarAlias id)
        | LocalDef (_,b,_) ->
          try project_variable (VarAlias id)
          with NotInvertibleUsingOurAlgorithm _ when imitate_defs ->
            imitate envk (EConstr.of_constr b))
    | LetIn (na,b,u,c) ->
        imitate envk (subst1 b c)
    | Evar (evk',args' as ev') ->
        if Evar.equal evk evk' then raise (OccurCheckIn (evd,rhs));
        (* At this point, we imitated a context say, C[ ], and virtually
           instantiated ?evk@{x₁..xn} with C[?evk''@{x₁..xn,y₁..yk}]
           for y₁..yk the spine of variables of C[ ], now facing the
           equation env, y₁...yk |- ?evk'@{args'} =?= ?evk''@{args,y1:=y1..yk:=yk} *)
        (* Assume evk' is defined in context x₁'..xk'.
           As a first step, we try to find a restriction ?evk'''@{xᵢ₁'..xᵢⱼ} of
           ?evk' and an instance args''' in the environment of ?evk such that
           env, y₁..yk |- args'''[args] = args' and thus such that
           env, y₁..yk |- ?evk'''@{args'''[args]} = ?evk''@{args,y1:=y1..yk:=yk} *)
        (* Note that we don't need to declare ?evk'' yet: it may remain virtual *)
        let aliases = lift_aliases k aliases in
        (try
          let ev = normalize_evar !evdref (evk,SList.Skip.map (lift k) argsv) in
          let evd,body = project_evar_on_evar false unify flags env' !evdref aliases k None ev' ev in
          evdref := evd;
          body
        with
        | EvarSolvedOnTheFly (evd,t) -> evdref:=evd; imitate envk t
        | CannotProject (evd,ev') ->
          if not !progress then
            raise (NotEnoughInformationEvarEvar t);
          (* We could not invert args' in terms of args, so we now make ?evk'' real *)
          let ty = get_type_of env' evd t in
          let (evd,evar'',ev'') =
             materialize_evar (evar_define unify flags ~choose) env' evd k ev ty in
          (* materialize_evar may instantiate ev' by another evar; adjust it *)
          let (evk',args' as ev') = normalize_evar evd ev' in
          let evd =
             (* Try now to invert args in terms of args' *)
            try
              let evd,body = project_evar_on_evar false unify flags env' evd aliases 0 None ev'' ev' in
              let evi = Evd.find_undefined evd evk' in
              let evd = Evd.define evk' body evd in
                check_evar_instance_evi unify flags env' evd evi body
            with
            | EvarSolvedOnTheFly _ -> assert false (* ev has no candidates *)
            | CannotProject (evd,ev'') ->
              (* ... or postpone the problem *)
              add_conv_oriented_pb (None,env',mkEvar ev'',mkEvar ev') evd in
          evdref := evd;
          evar'')
    | _ ->
        progress := true;
        match
          let c,args = decompose_app !evdref t in
          match EConstr.kind !evdref c with
          | Construct (cstr,u) when noccur_between !evdref 1 k t ->
            (* This is common case when inferring the return clause of match *)
            (* (currently rudimentary: we do not treat the case of multiple *)
            (*  possible inversions; we do not treat overlap with a possible *)
            (*  alternative inversion of the subterms of the constructor, etc)*)
            (match find_projectable_constructor env evd cstr k args cstr_subst with
             | _::_ as l -> Some (List.map mkVar l)
             | _ -> None)
          | _ -> None
        with
        | Some l ->
            let ty = get_type_of env' !evdref t in
            let candidates =
              try
                let t =
                  map_constr_with_full_binders env' !evdref (fun d (env,k) -> push_rel d env, k+1)
                    imitate envk t in
                (* Less dependent solutions come last *)
                l@[t]
              with e when CErrors.noncritical e -> l in
            (match candidates with
            | [x] -> x
            | _ ->
              let (evd,evar'',ev'') =
                materialize_evar (evar_define unify flags ~choose) env' !evdref k ev ty in
              evdref := restrict_evar evd (fst ev'') None (UpdateWith candidates);
              evar'')
        | None ->
           (* Evar/Rigid problem (or assimilated if not normal): we "imitate" *)
          map_constr_with_full_binders env' !evdref (fun d (env,k) -> push_rel d env, k+1)
                                        imitate envk t
  in

  let t' = imitate (env, 0) rhs in
  let body = if !progress then (recheck_applications unify flags (evar_env env evi) evdref t'; t') else t' in
  (!evdref,body)

(* [define] tries to solve the problem "?ev[args] = rhs" when "?ev" is
 * an (uninstantiated) evar such that "hyps |- ?ev : typ". Otherwise said,
 * [define] tries to find an instance lhs such that
 * "lhs [hyps:=args]" unifies to rhs. The term "lhs" must be closed in
 * context "hyps" and not referring to itself.
 * ev is assumed not to be frozen.
 *)

and evar_define unify flags ?(choose=false) ?(imitate_defs=true) env evd pbty (evk,argsv as ev) rhs =
  match EConstr.kind evd rhs with
  | Evar (evk2,argsv2 as ev2) ->
      if Evar.equal evk evk2 then
        solve_refl ~can_drop:choose
          (test_success unify) flags env evd pbty evk argsv argsv2
      else
        solve_evar_evar ~force:choose
          (evar_define unify flags) unify flags env evd pbty ev ev2
  | _ ->
  try solve_candidates unify flags env evd ev rhs
  with NoCandidates ->
  try
    let (evd',body) = invert_definition unify flags choose imitate_defs env evd pbty ev rhs in
    if occur_meta evd' body then raise MetaOccurInBodyInternal;
    (* invert_definition may have instantiate some evars of rhs with evk *)
    (* so we recheck acyclicity *)
    if occur_evar_upto_types evd' evk body then raise (OccurCheckIn (evd',body));
    (* needed only if an inferred type *)
    let evd', body = refresh_universes pbty env evd' body in
    instantiate_evar unify flags env evd' evk body
  with
    | NotEnoughInformationToProgress sols ->
        postpone_non_unique_projection env evd pbty ev sols rhs
    | NotEnoughInformationEvarEvar t ->
        add_conv_oriented_pb (pbty,env,mkEvar ev,t) evd
    | MorePreciseOccurCheckNeeeded ->
        add_conv_oriented_pb (pbty,env,mkEvar ev,rhs) evd
    | NotInvertibleUsingOurAlgorithm _ | MetaOccurInBodyInternal as e ->
        raise e
    | OccurCheckIn (evd,rhs) ->
        (* last chance: rhs actually reduces to ev *)
        let c = whd_all env evd rhs in
        match EConstr.kind evd c with
        | Evar (evk',argsv2) when Evar.equal evk evk' ->
            solve_refl (fun flags _b env sigma pb c c' -> is_fconv pb env sigma c c') flags
                       env evd pbty evk argsv argsv2
        | _ ->
            raise (OccurCheckIn (evd,rhs))

(* This code (i.e. solve_pb, etc.) takes a unification
 * problem, and tries to solve it. If it solves it, then it removes
 * all the conversion problems, and re-runs conversion on each one, in
 * the hopes that the new solution will aid in solving them.
 *
 * The kinds of problems it knows how to solve are those in which
 * the usable arguments of an existential var are all themselves
 * universal variables.
 * The solution to this problem is to do renaming for the Var's,
 * to make them match up with the Var's which are found in the
 * hyps of the existential, to do a "pop" for each Rel which is
 * not an argument of the existential, and a subst1 for each which
 * is, again, with the corresponding variable. This is done by
 * define
 *
 * Thus, we take the arguments of the existential which we are about
 * to assign, and zip them with the identifiers in the hypotheses.
 * Then, we process all the Var's in the arguments, and sort the
 * Rel's into ascending order.  Then, we just march up, doing
 * subst1's and pop's.
 *
 * NOTE: We can do this more efficiently for the relative arguments,
 * by building a long substituend by hand, but this is a pain in the
 * ass.
 *)

let status_changed evd lev (pbty,_,t1,t2) =
  (try Evar.Set.mem (head_evar evd t1) lev with NoHeadEvar -> false) ||
  (try Evar.Set.mem (head_evar evd t2) lev with NoHeadEvar -> false)

let reconsider_unif_constraints unify flags evd =
  let (evd,pbs) = extract_changed_conv_pbs evd (status_changed evd) in
  List.fold_left
    (fun p (pbty,env,t1,t2 as x) ->
       match p with
       | Success evd ->
           (match unify flags TermUnification env evd pbty t1 t2 with
           | Success _ as x -> x
           | UnifFailure (i,e) -> UnifFailure (i,CannotSolveConstraint (x,e)))
       | UnifFailure _ as x -> x)
    (Success evd)
    pbs

(* Tries to solve problem t1 = t2.
 * Precondition: t1 is an uninstantiated evar
 * Returns an optional list of evars that were instantiated, or None
 * if the problem couldn't be solved. *)

(* Rq: incomplete algorithm if pbty = CONV_X_LEQ ! *)
let solve_simple_eqn unify flags ?(choose=false) ?(imitate_defs=true)
                     env evd (pbty,(evk1,args1 as ev1),t2) =
  try
    let t2 = whd_betaiota env evd t2 in (* includes whd_evar *)
    let evd = evar_define unify flags ~choose ~imitate_defs env evd pbty ev1 t2 in
      reconsider_unif_constraints unify flags evd
  with
    | NotInvertibleUsingOurAlgorithm t ->
        UnifFailure (evd,NotClean (ev1,env,t))
    | OccurCheckIn (evd,rhs) ->
        UnifFailure (evd,OccurCheck (evk1,rhs))
    | MetaOccurInBodyInternal ->
        UnifFailure (evd,MetaOccurInBody evk1)
    | IllTypedInstance (env,evd,t,u) ->
        UnifFailure (evd,InstanceNotSameType (evk1,env,t,u))
    | IllTypedInstanceFun (env,evd,f,u) ->
        UnifFailure (evd,InstanceNotFunctionalType (evk1,env,f,u))
    | IncompatibleCandidates t ->
        UnifFailure (evd,IncompatibleInstances (env,ev1,t,t2))

