(************************************************************************)
(*         *      The Rocq Prover / The Rocq 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 CErrors
open Util
open Names
open Univ
open Sorts
open UVars

module PContextSet = struct
  open PConstraints
  type t = Level.Set.t constrained

  let empty = (Level.Set.empty, empty)
  let is_empty (univs, cst) = Level.Set.is_empty univs && is_empty cst

  let union (univs, cst as x) (univs', cst' as y) =
    if x == y then x
    else Level.Set.union univs univs', union cst cst'

  let add_level u (univs, cst) =
    Level.Set.add u univs, cst

  let pr printer (univs, cst) =
    UnivGen.pr_sort_context printer ((Sorts.QVar.Set.empty, univs), cst)

  let univ_context_set (uvars, (_, uctx)) = (uvars, uctx)
  let univ_constraints (_, (_,csts)) = csts
  let levels (univs, _cst) = univs

end

let template_default_univs = Summary.ref ~name:"template default univs" Univ.Level.Set.empty

let cache_template_default_univs us =
  let open Summary.Ref in
  template_default_univs := Univ.Level.Set.union !template_default_univs us

let template_default_univs_obj =
  Libobject.declare_object {
    (Libobject.default_object "template default univs") with
    cache_function = cache_template_default_univs;
    load_function = (fun _ us -> cache_template_default_univs us);
    discharge_function = (fun x -> Some x);
    classify_function = (fun _ -> Escape);
  }

let add_template_default_univs env kn =
  match (Environ.lookup_mind kn env).mind_template with
  | None -> ()
  | Some template ->
    let _, us = UVars.Instance.levels template.template_defaults in
    Lib.add_leaf (template_default_univs_obj us)

let template_default_univs () =
  let open Summary.Ref in
  !template_default_univs

module UnivFlex = UnivFlex

type universes_entry =
| Monomorphic_entry of Univ.ContextSet.t
| Polymorphic_entry of UVars.UContext.t

module UNameMap = Id.Map

type uinfo = {
  uname : Id.t option;
  uloc : Loc.t option;
}

open Quality

exception SortInconsistency of UGraph.univ_inconsistency

let sort_inconsistency ?explain cst l r =
  let explain = Option.map (fun p -> UGraph.Other p) explain in
  raise (SortInconsistency (None, (cst, l, r, explain)))

let univ_inconsistency ?explain cst l r =
  let explain = Option.map (fun p -> UGraph.Other p) explain in
  raise (UGraph.UniverseInconsistency (None, (cst, l, r, explain)))

module QSet = QVar.Set
module QMap = QVar.Map

type above_prop = AboveProp | BelowType

module QState : sig
  type t
  type elt = QVar.t
  val empty : t
  val union : fail:(t -> Quality.t -> Quality.t -> t) -> t -> t -> t
  val add : check_fresh:bool -> rigid:bool -> elt -> t -> t
  val repr : elt -> t -> Quality.t
  val is_rigid : t -> QVar.t -> bool
  val is_above_prop : t -> QVar.t -> bool
  val set_above_prop : QVar.t -> above_prop -> t -> t option
  val unify_quality : fail:(unit -> t) -> Conversion.conv_pb -> Quality.t -> Quality.t -> t -> t
  val undefined : t -> QVar.Set.t
  val collapse : ?except:QVar.Set.t -> only_above_prop:bool -> t -> t
  val pr : Sorts.Quality.printer -> (QVar.t -> Id.t option) -> t -> Pp.t
  val of_elims : QGraph.t -> t
  val elims : t -> QGraph.t
  val set_elims : QGraph.t -> t -> t
  val initial_elims : t -> QGraph.t
  val merge_constraints : (QGraph.t -> QGraph.t) -> t -> t
  val normalize_elim_constraints : t -> ElimConstraints.t -> ElimConstraints.t
end =
struct

type node =
| Equiv of Quality.t
| Canonical of { rigid : bool }
(** Rigid variables may not be set to another *)

type t = {
  qmap : node QMap.t;
  (* TODO: use a persistent union-find structure *)
  above_prop : above_prop QMap.t;
  (** Set for quality variables known to be either in Prop or Type.
      If q ∈ above_prop then it must map to None in qmap. *)
  elims : QGraph.t;
  (** Elimination graph for quality variables. *)
  initial_elims : QGraph.t;
  (** Keep the qvar domain without any constraints to optimize computation. *)
}

type elt = QVar.t

let empty = { qmap = QMap.empty; above_prop = QMap.empty;
              elims = QGraph.initial_graph; initial_elims = QGraph.initial_graph }

let rec repr q m = match QMap.find q m.qmap with
| Canonical _ -> QVar q
| Equiv (QVar q) -> repr q m
| Equiv (QConstant _ | QGlobal _ as q) -> q
| exception Not_found -> QVar q

type repr =
| ReprConstant of Quality.constant
| ReprGlobal of QGlobal.t
| ReprVar of QVar.t * bool

let rec repr_node_qvar q m = match QMap.find q m.qmap with
| Canonical { rigid } -> ReprVar (q, rigid)
| Equiv q -> repr_node q m
| exception Not_found -> ReprVar (q, true) (* a bit dubious but missing variables are considered rigid *)

and repr_node q m = match q with
| QVar q -> repr_node_qvar q m
| (QConstant qc) -> ReprConstant qc
| (QGlobal q) -> ReprGlobal q

let is_above_prop m q = QMap.mem q m.above_prop

let above_prop m q = QMap.find_opt q m.above_prop

let eliminates_to_prop m q =
  QGraph.eliminates_to_prop m.elims (QVar q)

let is_rigid m q = match repr_node_qvar q m with
| ReprVar (_, rigid) -> rigid
| ReprConstant _ | ReprGlobal _ -> true

let set q qv m =
  let q = repr_node_qvar q m in
  let q, rigid = match q with
    | ReprVar (q, rigid) -> q, rigid
    | ReprConstant _ | ReprGlobal _ -> assert false
  in
  let qv = repr_node qv m in
  let enforce_eq q1 q2 g =
    let ans = QGraph.enforce_eliminates_to q1 q2 (QGraph.enforce_eliminates_to q2 q1 g) in
    let () = QGraph.check_rigid_paths ans in
    ans
  in
  match qv with
  | ReprVar (qv, _qvrigd) ->
    if QVar.equal q qv then Some m
    else if rigid then None
    else
      let above_prop =
        match above_prop m q with
        | None -> m.above_prop
        | Some above -> QMap.add qv above (QMap.remove q m.above_prop)
      in
      Some { m with
             qmap = QMap.add q (Equiv (QVar qv)) m.qmap; above_prop;
             elims = enforce_eq (QVar qv) (QVar q) m.elims; }
  | ReprConstant qc ->
    if qc == QSProp && (is_above_prop m q || eliminates_to_prop m q) then None
    else if rigid then None
    else
      let qv = QConstant qc in
      Some { m with qmap = QMap.add q (Equiv qv) m.qmap;
                    above_prop = QMap.remove q m.above_prop;
                    elims = enforce_eq qv (QVar q) m.elims }
  | ReprGlobal qg ->
    if is_above_prop m q then None
    else if rigid then None
    else
      let qv = QGlobal qg in
      Some { m with qmap = QMap.add q (Equiv qv) m.qmap;
                    above_prop = m.above_prop;
                    elims = enforce_eq qv (QVar q) m.elims }

let set_above_prop q above m =
  let q = repr_node_qvar q m in
  let q, rigid = match q with ReprVar (q, rigid) -> q, rigid | ReprConstant _ | ReprGlobal _ -> assert false in
  if rigid then None
  else
    let above_prop =
      QMap.update q (fun prev ->
          match above, prev with
          | _, Some BelowType -> Some BelowType
          | (AboveProp|BelowType), _ -> Some above)
        m.above_prop
    in
    Some { m with above_prop }

let unify_quality ~fail c q1 q2 local = match q1, q2 with
| QConstant QType, QConstant QType
| QConstant QProp, QConstant QProp
| QConstant QSProp, QConstant QSProp -> local
| QGlobal q1, QGlobal q2 -> if QGlobal.equal q1 q2 then local else fail ()
| QConstant QProp, QVar q when c == Conversion.CUMUL ->
  begin match set_above_prop q AboveProp local with
  | Some local -> local
  | None -> fail ()
  end
| QVar qv1, QVar qv2 -> begin match set qv1 q2 local with
    | Some local -> local
    | None -> match set qv2 q1 local with
      | Some local -> local
      | None -> fail ()
  end
(* TODO BelowType *)
| QVar q, (QConstant (QType | QProp | QSProp) | QGlobal _ as qv)
| (QConstant (QType | QProp | QSProp) | QGlobal _ as qv), QVar q ->
  begin match set q qv local with
  | Some local -> local
  | None -> fail ()
  end
| QGlobal _, QConstant _| QConstant _, QGlobal _ -> fail ()
| (QConstant QType, QConstant (QProp | QSProp)) -> fail ()
| (QConstant QProp, QConstant QType) ->
  begin match c with
  | CONV -> fail ()
  | CUMUL -> local
  end
| (QConstant QSProp, QConstant (QType | QProp)) -> fail ()
| (QConstant QProp, QConstant QSProp) -> fail ()

let nf_quality m = function
  | QConstant _ | QGlobal _ as q -> q
  | QVar q -> repr q m

let add_qvars m qmap qs =
  let g = m.initial_elims in
  let filter v = match QMap.find_opt v qmap with
  | None | Some (Canonical _) -> true
  | Some (Equiv _) -> false
  in
  (* Here, we filter instead of enforcing equality due to the collapse:
     simply enforcing equality may lead to inconsistencies after it *)
  let qs = QVar.Set.filter filter qs in
  let fold v g = try QGraph.add_quality (QVar v) g with QGraph.AlreadyDeclared -> g in
  QVar.Set.fold fold qs g

let union ~fail s1 s2 =
  let extra = ref [] in
  let qmap = QMap.union (fun qk q1 q2 ->
      match q1, q2 with
      | Equiv q, (Canonical {rigid}) | (Canonical {rigid}), Equiv q ->
        assert (not rigid);
        Some (Equiv q)
      | Canonical { rigid = r1 }, Canonical { rigid = r2 } ->
        assert (Bool.equal r1 r2);
        Some (Canonical { rigid = r1 })
      | Equiv q1, Equiv q2 ->
        let () = if not (Quality.equal q1 q2) then extra := (q1,q2) :: !extra in
        Some (Equiv q1))
      s1.qmap s2.qmap
  in
  let extra = !extra in
  let qs = QVar.Set.union (QGraph.qvar_domain s1.elims) (QGraph.qvar_domain s2.elims) in
  let filter_above v _ = match QMap.find_opt v qmap with
  | None | Some (Canonical _) -> true
  | Some (Equiv _) -> false
  in
  let above_union _ a b = match a, b with
    | AboveProp, AboveProp -> Some AboveProp
    | BelowType, _ | _, BelowType -> Some BelowType
  in
  let above_prop = QMap.filter filter_above @@ QMap.union above_union s1.above_prop s2.above_prop in
  let elims = add_qvars s2 qmap qs in
  let s = { qmap; above_prop;
            elims; initial_elims = elims } in
  List.fold_left (fun s (q1,q2) ->
      let q1 = nf_quality s q1 and q2 = nf_quality s q2 in
      unify_quality ~fail:(fun () -> fail s q1 q2) CONV q1 q2 s)
    s
    extra

let add ~check_fresh ~rigid q m =
  if check_fresh then assert (not (QMap.mem q m.qmap));
  let add_quality g =
    try QGraph.add_quality (QVar q) g
    with QGraph.AlreadyDeclared as e -> if check_fresh then raise e else g
  in
  { qmap = QMap.add q (Canonical { rigid }) m.qmap;
    above_prop = m.above_prop;
    elims = add_quality m.elims;
    initial_elims = add_quality m.initial_elims }

let of_elims elims =
  { empty with elims; initial_elims = elims }

(* XXX what about qvars in the elimination graph? *)
let undefined m =
  let filter _ v = match v with
  | Canonical _ -> true
  | Equiv _ -> false
  in
  let mq = QMap.filter filter m.qmap in
  QMap.domain mq

let collapse ?(except=QSet.empty) ~only_above_prop m =
  let free_qualities = QMap.fold (fun q v fqs ->
      match v with
      | Equiv _ -> fqs
      | Canonical _ -> QSet.add q fqs)
      m.qmap QSet.empty
  in
  let dominates_above_prop q q' =
    not (QVar.equal q q') && QGraph.eliminates_to m.elims (QVar q) (QVar q') && not (QMap.mem q m.above_prop)
  in
  QMap.fold (fun q v m ->
      match v with
      | Equiv _ -> m
      | Canonical { rigid } ->
        if rigid || QSet.mem q except then m
        (* This check is necessary because there is a particular scenario where we could end up
           with an unsatisfied elimination constraint or an unnecessary elaborated elimination constraint to Type
           (or some inexistent sort variable); if we simply defaulted sort variables to Type, as before.
           This comes from a weird interaction with "above Prop". The scenario is:
           - An unbound sort variable β might be set to be above Prop during unification, which in practice
             should be equal to Prop.
           - A rigid sort s eliminates to Prop explicitly (and β, since they are supposed to be equal)
           - Collapsing β to Type means that now sort s eliminates to Type, but this is an undeclared constraint,
             and therefore the declaration fails.

           This check is therefore simply finding if the sort variable above Prop is dominated by another one.
           If so, the sort variable collapses to Prop, otherwise to Type (if collapsing is enabled), or we keep it.
           TODO not sure if still needed with BelowType
        *)
        else match QMap.find_opt q m.above_prop with
          | Some BelowType ->
            if QSet.exists (fun q' -> dominates_above_prop q' q) free_qualities then
              Option.get (set q qprop m)
            else Option.get (set q qtype m)
          | Some AboveProp -> Option.get (set q qprop m)
          | None -> if not only_above_prop then Option.get (set q qtype m) else m)
    m.qmap m

let pr prqvar local_name ({ qmap; elims } as m) =
  let open Pp in
  (* Print the "body" of the QVar, e.g. "α1 := Type", "α2 >= Prop" *)
  let prbody u = function
  | Canonical { rigid } ->
    begin match above_prop m u with
    | Some AboveProp -> str " >= Prop"
    | Some BelowType -> str " <= Type"
    | None ->
      if rigid then
        str " (rigid)"
      else mt ()
    end
  | Equiv q ->
    let q = Quality.pr prqvar q in
    str " := " ++ q
  in
  (* Print the "name" (given by the user) of the Qvar, e.g. "(named s)" *)
  let prqvar_name q = match local_name q with
  | None -> mt ()
  | Some qid -> str " (named " ++ Id.print qid ++ str ")"
  in
  let prqvar_full (q1, q2) = QVar.raw_pr q1 ++ prbody q1 q2 ++ prqvar_name q1 in
  hov 0 (prlist_with_sep fnl prqvar_full (QMap.bindings qmap) ++
    str " |=" ++ brk (1, 2) ++ hov 0 (QGraph.pr_qualities prqvar elims))

let elims m = m.elims

let set_elims elims m = { m with elims }

let initial_elims m = m.initial_elims

let merge_constraints f m =
  { m with elims = f m.elims }

let normalize_elim_constraints m cstrs =
  let subst q = match q with
    | QConstant _ | QGlobal _ -> q
    | QVar qv -> repr qv m
  in
  let is_instantiated q = is_qconst q || is_qglobal q in
  let can_drop (q1,_,q2) = not (is_instantiated q1 && is_instantiated q2) in
  let subst_cst (q1,c,q2) = (subst q1,c,subst q2) in
  let cstrs = ElimConstraints.map subst_cst cstrs in
  let cstrs = ElimConstraints.filter can_drop cstrs in
  ElimConstraints.filter (fun (q1, _, q2) -> not @@ Quality.equal q1 q2) cstrs
end

module UPairSet = UnivMinim.UPairSet

type univ_names = UnivNames.universe_binders * (uinfo QVar.Map.t * uinfo Level.Map.t)

(* 2nd part used to check consistency on the fly. *)
type t =
 { names : univ_names; (** Printing/location information *)
   local : PContextSet.t; (** The local graph of universes (variables and constraints) *)
   univ_variables : UnivFlex.t;
   (** The local universes that are unification variables *)
   sort_variables : QState.t;
   (** Local quality variables. *)
   universes : UGraph.t; (** The current graph extended with the local constraints *)
   initial_universes : UGraph.t; (** The graph at the creation of the evar_map + local universes
                                     (but not local constraints) *)
   minim_extra : UnivMinim.extra;
 }

let empty =
  { names = UnivNames.empty_binders, (QMap.empty, Level.Map.empty);
    local = PContextSet.empty;
    univ_variables = UnivFlex.empty;
    sort_variables = QState.empty;
    universes = UGraph.initial_universes;
    initial_universes = UGraph.initial_universes;
    minim_extra = UnivMinim.empty_extra; }

let is_empty uctx =
  PContextSet.is_empty uctx.local &&
  UnivFlex.is_empty uctx.univ_variables

let id_of_level uctx l =
  try (Level.Map.find l (snd (snd uctx.names))).uname
  with Not_found -> None

let loc_of_level uctx l =
  try (Level.Map.find l (snd (snd uctx.names))).uloc
  with Not_found -> None

let id_of_qvar uctx l =
  try (QVar.Map.find l (fst (snd uctx.names))).uname
  with Not_found -> None

let is_rigid_qvar uctx q = QState.is_rigid uctx.sort_variables q

let get_uname info = match info.uname with
| None -> raise Not_found
| Some id -> id

let qualid_of_qvar_names (bind, (qrev,_)) l =
  try Some (Libnames.qualid_of_ident (get_uname (QVar.Map.find l qrev)))
  with Not_found ->
    UnivNames.qualid_of_quality bind (QVar l)

let qualid_of_level_names (bind, (_,urev)) l =
  try Some (Libnames.qualid_of_ident (get_uname (Level.Map.find l urev)))
  with Not_found ->
    UnivNames.qualid_of_level bind l

let qualid_of_level uctx l = qualid_of_level_names uctx.names l

let pr_uctx_qvar_names names l =
  match qualid_of_qvar_names names l with
  | Some qid -> Libnames.pr_qualid qid
  | None -> QVar.raw_pr l

let quality_printer_names names = {
  Sorts.Quality.prvar = pr_uctx_qvar_names names;
  prglobal = (UnivNames.quality_printer (fst names)).prglobal;
}

let quality_printer uctx = quality_printer_names uctx.names

let pr_uctx_level_names names l =
  match qualid_of_level_names names l with
  | Some qid -> Libnames.pr_qualid qid
  | None -> Level.raw_pr l

let sort_printer_names names = {
  Sorts.prq = quality_printer_names names;
  pru = pr_uctx_level_names names;
}

let sort_printer uctx = sort_printer_names uctx.names

let pr_uctx_level uctx l = pr_uctx_level_names uctx.names l

let pr_uctx_qglobal uctx q =
  UnivNames.pr_quality_with_global_universes ~binders:(fst uctx.names) (QGlobal q)

let pr_uctx_qvar uctx l = pr_uctx_qvar_names uctx.names l

let merge_univ_constraints uctx cstrs g =
  try UGraph.merge_constraints cstrs g
  with UGraph.UniverseInconsistency (_, i) ->
    raise (UGraph.UniverseInconsistency (Some (sort_printer uctx), i))

type constraint_source =
| Internal
| Rigid
| Static

let merge_elim_constraints ?(src = Internal) uctx cstrs g =
  try
    let g = QGraph.merge_constraints cstrs g in
    match src with
    | Static -> g
    | Internal ->
      let () = if not (ElimConstraints.is_empty cstrs) then QGraph.check_rigid_paths g in
      g
    | Rigid ->
      let fold (q1, _, q2) accu = QGraph.add_rigid_path q1 q2 accu in
      Sorts.ElimConstraints.fold fold cstrs g
  with QGraph.(EliminationError (QualityInconsistency (_, i))) ->
    raise (QGraph.(EliminationError (QualityInconsistency (Some (quality_printer uctx), i))))

let uname_union s t =
  if s == t then s
  else
    UNameMap.merge (fun k l r ->
        match l, r with
        | Some _, _ -> l
        | _, _ -> r) s t

let names_union ((qbind,ubind),(qrev,urev)) ((qbind',ubind'),(qrev',urev')) =
  let qbind = uname_union qbind qbind'
  and ubind = uname_union ubind ubind'
  and qrev = QVar.Map.union (fun _ l _ -> Some l) qrev qrev'
  and urev = Level.Map.lunion urev urev' in
  ((qbind,ubind),(qrev,urev))

let union uctx uctx' =
  if uctx == uctx' then uctx
  else if is_empty uctx' then uctx
  else
    let local = PContextSet.union uctx.local uctx'.local in
    let names = names_union uctx.names uctx'.names in
    let newus = Level.Set.diff (PContextSet.levels uctx'.local)
                               (PContextSet.levels uctx.local) in
    let newus = Level.Set.diff newus (UnivFlex.domain uctx.univ_variables) in
    let extra = UnivMinim.extra_union uctx.minim_extra uctx'.minim_extra in
    let declarenew g =
      Level.Set.fold (fun u g -> UGraph.add_universe u ~strict:false g) newus g
    in
    let fail_union s q1 q2 =
      if QGraph.ignore_constraints (QState.elims uctx.sort_variables) then s else
      CErrors.user_err
        Pp.(str "Could not merge universe contexts: could not unify" ++ spc() ++
           Quality.raw_pr q1 ++ strbrk " and " ++ Quality.raw_pr q2 ++ str ".")
    in
      { names;
        local = local;
        univ_variables =
          UnivFlex.biased_union uctx.univ_variables uctx'.univ_variables;
        (* FIXME: merge constraints from contextset *)
        sort_variables = QState.union ~fail:fail_union uctx.sort_variables uctx'.sort_variables;
        initial_universes = declarenew uctx.initial_universes;
        universes =
          (if local == uctx.local then uctx.universes
           else
             let cstrsr = PContextSet.univ_constraints uctx'.local in
             merge_univ_constraints uctx cstrsr (declarenew uctx.universes));
        minim_extra = extra}

let context_set uctx = uctx.local

let universe_context_set uctx =
  let us, (_, ucst) = uctx.local in
  us, ucst

let sort_context_set uctx =
  let us, csts = uctx.local in
  (QState.undefined uctx.sort_variables, us), csts

let constraints uctx = snd uctx.local

let compute_instance_binders uctx inst =
  let (qrev, urev) = snd uctx.names in
  let qinst, uinst = Instance.to_array inst in
  let qmap = function
    | QVar q ->
      begin try Name (get_uname (QVar.Map.find q qrev))
      with Not_found -> Anonymous
      end
    | QConstant _ | QGlobal _ -> assert false
  in
  let umap lvl =
    try Name (get_uname (Level.Map.find lvl urev))
    with Not_found -> Anonymous
  in
  {quals = Array.map qmap qinst; univs =  Array.map umap uinst}

let context uctx =
  let qvars = QState.undefined uctx.sort_variables in
  let (uvars, (qcst, ucst)) = uctx.local in
  UContext.of_context_set (compute_instance_binders uctx) ((qvars, qcst), (uvars, ucst))

type named_universes_entry = universes_entry * UnivNames.universe_binders

let check_mono_sort_constraints uctx =
  let (uvar, (qcst, ucst)) = uctx.local in
  (* This looks very stringent but it passes nonetheless all the tests? *)
  let () = assert (Sorts.ElimConstraints.is_empty qcst) in
  (uvar, ucst)

let univ_entry ~poly uctx =
  let (binders, _) = uctx.names in
  let entry =
    if PolyFlags.univ_poly poly then Polymorphic_entry (context uctx)
    else
      let uctx = check_mono_sort_constraints uctx in
      Monomorphic_entry uctx
  in
  entry, binders

type universe_opt_subst = UnivFlex.t

let subst uctx = uctx.univ_variables
let ugraph uctx = uctx.universes

let elim_graph uctx = QState.elims uctx.sort_variables
let initial_elim_graph uctx = QState.initial_elims uctx.sort_variables

let is_above_prop uctx qv = QState.is_above_prop uctx.sort_variables qv

let is_algebraic l uctx = UnivFlex.is_algebraic l uctx.univ_variables

(** Deprecated *)
let of_names (ubind,(revqbind,revubind)) =
  let revqbind = QVar.Map.map (fun id -> { uname = Some id; uloc = None }) revqbind in
  let revubind = Level.Map.map (fun id -> { uname = Some id; uloc = None }) revubind in
  let qgraph = QVar.Map.fold (fun v _ -> QGraph.add_quality (QVar v)) revqbind QGraph.initial_graph in
  { empty with names = (ubind,(revqbind,revubind));
               sort_variables = QState.of_elims qgraph; }

let universe_of_name uctx s =
  UNameMap.find s (snd (fst uctx.names))

let quality_of_name uctx s =
  Id.Map.find s (fst (fst uctx.names))

let name_level level id uctx =
  let ((qbind,ubind),(qrev,urev)) = uctx.names in
  assert(not(Id.Map.mem id ubind));
  let ubind = Id.Map.add id level ubind in
  let urev = Level.Map.add level { uname = Some id; uloc = None } urev in
  { uctx with names = ((qbind,ubind),(qrev,urev)) }

let universe_binders uctx =
  let named, _ = uctx.names in
  named

let nf_qvar uctx q = QState.repr q uctx.sort_variables

let instantiate_variable l (b : Universe.t) v =
  v := UnivFlex.define l b !v

exception UniversesDiffer

let { Goptions.get = weak_constraints } =
  Goptions.declare_bool_option_and_ref
    ~key:["Cumulativity";"Weak";"Constraints"]
    ~value:true
    ()

let level_inconsistency cst l r =
  let mk u = Sorts.sort_of_univ @@ Universe.make u in
  raise (UGraph.UniverseInconsistency (None, (cst, mk l, mk r, None)))

let nf_universe uctx u =
  UnivSubst.(subst_univs_universe (UnivFlex.normalize_univ_variable uctx.univ_variables)) u

let nf_level uctx u =
  UnivSubst.(level_subst_of (UnivFlex.normalize_univ_variable uctx.univ_variables)) u

let nf_instance uctx u = Instance.subst_fn (nf_qvar uctx, nf_level uctx) u

let nf_quality uctx q = Quality.subst (nf_qvar uctx) q

let nf_sort uctx s =
  let normalize u = nf_universe uctx u in
  let qnormalize q = QState.repr q uctx.sort_variables in
  Sorts.subst_fn (qnormalize, normalize) s

let nf_relevance uctx r = match r with
| Relevant | Irrelevant -> r
| RelevanceVar q ->
  match nf_qvar uctx q with
  | QConstant QSProp -> Sorts.Irrelevant
  | QConstant QProp | QConstant QType | QGlobal _ -> Sorts.Relevant
  | QVar q' ->
    (* XXX currently not used in nf_evars_and_universes_opt_subst
       does it matter? *)
    if QState.is_above_prop uctx.sort_variables q' then Relevant
    else if QVar.equal q q' then r
    else Sorts.RelevanceVar q'

let nf_universes uctx c =
  let lsubst = uctx.univ_variables in
  let nf_univ u = UnivFlex.normalize_univ_variable lsubst u in
  let rec self () c = match Constr.kind c with
  | Evar (evk, args) ->
    let args' = SList.Smart.map (self ()) args in
    if args == args' then c else Constr.mkEvar (evk, args')
  | _ -> UnivSubst.map_universes_opt_subst_with_binders ignore self (nf_relevance uctx) (nf_qvar uctx) nf_univ () c
  in
  self ()  c

type small_universe = USet | UProp | USProp

let is_uset = function USet -> true | UProp | USProp -> false

type sort_classification =
| USmall of small_universe (* Set, Prop or SProp *)
| ULevel of Level.t (* Var or Global *)
| UMax of Universe.t * Level.Set.t (* Max of Set, Var, Global without increments *)
| UAlgebraic of Universe.t (* Arbitrary algebraic expression *)

let classify s = match s with
| Prop -> USmall UProp
| SProp -> USmall USProp
| Set -> USmall USet
| Type u | GSort (_, u) | VSort (_, u) ->
  if Universe.is_levels u then match Universe.level u with
  | None -> UMax (u, Universe.levels u)
  | Some u -> ULevel u
  else UAlgebraic u

type local = {
  local_cst : PConstraints.t;
  local_above_prop : Level.Set.t;
  local_weak : UPairSet.t;
  local_sorts : QState.t;
}

let add_univ_local cst local =
  { local with local_cst = PConstraints.add_univ cst local.local_cst }

(* Constraint with algebraic on the left and a single level on the right *)
let enforce_leq_up u v local =
  let elim_csts = PConstraints.qualities local.local_cst in
  let univ_csts = UnivSubst.enforce_leq u (Universe.make v) @@
                    PConstraints.univs local.local_cst in
  { local with local_cst = PConstraints.make elim_csts univ_csts }

let get_constraint = function
| Conversion.CONV -> UnivConstraint.Eq
| Conversion.CUMUL -> UnivConstraint.Le

let warn_template =
  CWarnings.create_warning ~from:[CWarnings.CoreCategories.fragile] ~default:Disabled ~name:"bad-template-constraint" ()

let do_warn_template = CWarnings.create_in warn_template
    Pp.(fun (uctx,csts) ->
        str "Adding constraints involving global template univs:" ++ spc() ++
        UnivConstraints.pr (pr_uctx_level uctx) csts )

let warn_template uctx csts =
  match CWarnings.warning_status warn_template with
  | Disabled -> ()
  | Enabled | AsError ->
  let is_template u = Level.Set.mem u (template_default_univs()) in
  let csts = UnivConstraints.filter (fun (u,_,v as cst) ->
      not (Level.is_set u) && not (Level.is_set v) &&
      (is_template u || is_template v) &&
      not (UGraph.check_constraint uctx.universes cst)) csts in
  if not @@ UnivConstraints.is_empty csts then
    do_warn_template (uctx,csts)

let unify_quality c s1 s2 l =
  let fail () =
    if QGraph.ignore_constraints (QState.elims l.local_sorts) then l.local_sorts else
    sort_inconsistency (get_constraint c) s1 s2
  in
  { l with
    local_sorts = QState.unify_quality ~fail
        c (Sorts.quality s1) (Sorts.quality s2) l.local_sorts;
  }

let process_constraints uctx cstrs =
  let open UnivSubst in
  let open UnivProblem in
  let univs = uctx.universes in
  let vars = ref uctx.univ_variables in
  let normalize u = UnivFlex.normalize_univ_variable !vars u in
  let qnormalize sorts q = QState.repr q sorts in
  let normalize_sort sorts s =
    Sorts.subst_fn ((qnormalize sorts), subst_univs_universe normalize) s
  in
  let nf_constraint sorts = function
    | QElimTo (a, b) -> QElimTo (Quality.subst (qnormalize sorts) a, Quality.subst (qnormalize sorts) b)
    | QLeq (a, b) -> QLeq (Quality.subst (qnormalize sorts) a, Quality.subst (qnormalize sorts) b)
    | QEq (a, b) -> QEq (Quality.subst (qnormalize sorts) a, Quality.subst (qnormalize sorts) b)
    | ULub (u, v) -> ULub (level_subst_of normalize u, level_subst_of normalize v)
    | UWeak (u, v) -> UWeak (level_subst_of normalize u, level_subst_of normalize v)
    | UEq (u, v) -> UEq (normalize_sort sorts u, normalize_sort sorts v)
    | ULe (u, v) -> ULe (normalize_sort sorts u, normalize_sort sorts v)
  in
  let is_local l = UnivFlex.mem l !vars in
  let equalize_small l s local =
    let ls = match l with
    | USProp -> sprop
    | UProp -> prop
    | USet -> set
    in
    if UGraph.check_eq_sort Sorts.Quality.equal univs ls s then local
    else if is_uset l then match classify s with
    | USmall _ -> univ_inconsistency Eq set s
    | ULevel r ->
      if is_local r then
        let () = instantiate_variable r Universe.type0 vars in
        add_univ_local (Level.set, Eq, r) local
      else
        univ_inconsistency Eq set s
    | UMax (u, _)| UAlgebraic u ->
      if univ_level_mem Level.set u then
        let inst = univ_level_rem Level.set u u in
        enforce_leq_up inst Level.set local
      else
        univ_inconsistency Eq ls s
    else univ_inconsistency Eq ls s
  in
  let equalize_variables fo l' r' local =
    if Level.equal l' r' then local
    else
      let () =
        if is_local l' then
          instantiate_variable l' (Universe.make r') vars
        else if is_local r' then
          instantiate_variable r' (Universe.make l') vars
        else if not (UnivProblem.check_eq_level univs l' r') then
          (* Two rigid/global levels, none of them being local,
              one of them being Prop/Set, disallow *)
          if Level.is_set l' || Level.is_set r' then
            level_inconsistency Eq l' r'
          else if fo then
            raise UniversesDiffer
      in
      add_univ_local (l', Eq, r') local
  in
  let equalize_algebraic l ru local =
    let alg = UnivFlex.is_algebraic l uctx.univ_variables in
    let inst = univ_level_rem l ru ru in
    if alg && not (Level.Set.mem l (Universe.levels inst)) then
      let () = instantiate_variable l inst vars in
      local
    else
      if univ_level_mem l ru then
        enforce_leq_up inst l local
      else univ_inconsistency Eq (sort_of_univ (Universe.make l)) (sort_of_univ ru)
  in
  let equalize_universes l r local = match classify l, classify r with
  | USmall l', (USmall _ | ULevel _ | UMax _ | UAlgebraic _) ->
    equalize_small l' r local
  | (ULevel _ | UMax _ | UAlgebraic _), USmall r' ->
    equalize_small r' l local
  | ULevel l', ULevel r' ->
    equalize_variables false l' r' local
  | ULevel l', (UAlgebraic r | UMax (r, _)) | (UAlgebraic r | UMax (r, _)), ULevel l' ->
    equalize_algebraic l' r local
  | (UAlgebraic _ | UMax _), (UAlgebraic _ | UMax _) ->
    (* both are algebraic *)
    if UGraph.check_eq_sort Sorts.Quality.equal univs l r then local
    else univ_inconsistency Eq l r
  in
  let unify_universes cst local =
    let cst = nf_constraint local.local_sorts cst in
    if UnivProblem.is_trivial cst then local
    else
      (* TODO sort_inconsistency should be able to handle raw
         qualities instead of having to make a dummy sort *)
      let mk q = Sorts.make q Universe.type0 in
      match cst with
    | QEq (a, b) -> unify_quality CONV (mk a) (mk b) local
    | QLeq (a, b) -> unify_quality CUMUL (mk a) (mk b) local
    | QElimTo (a, b) -> { local with local_cst = PConstraints.add_quality (a, ElimTo, b) local.local_cst }
    | ULe (l, r) ->
      let local = unify_quality CUMUL l r local in
      let l = normalize_sort local.local_sorts l in
      let r = normalize_sort local.local_sorts r in
      begin match classify r with
      | UAlgebraic _ | UMax _ ->
        if UGraph.check_leq_sort Sorts.Quality.equal univs l r then local
        else
          univ_inconsistency Le l r
            ~explain:(Pp.str "(cannot handle algebraic on the right)")
      | USmall r' ->
        (* Invariant: there are no universes u <= Set in the graph. Except for
           template levels, Set <= u anyways. Otherwise, for template
           levels, any constraint u <= Set is turned into u := Set. *)
        if UGraph.type_in_type univs then local
        else begin match classify l with
        | UAlgebraic _ ->
          (* l contains a +1 and r=r' small so l <= r impossible *)
          univ_inconsistency Le l r
        | USmall l' ->
          if UGraph.check_leq_sort Sorts.Quality.equal univs l r then local
          else univ_inconsistency Le l r
        | ULevel l' ->
          if is_uset r' && is_local l' then
            (* Unbounded universe constrained from above, we equalize it *)
            let () = instantiate_variable l' Universe.type0 vars in
            add_univ_local (l', Eq, Level.set) local
          else
            univ_inconsistency Le l r
        | UMax (_, levels) ->
          if is_uset r' then
            let fold l' local =
              let l = sort_of_univ @@ Universe.make l' in
              if Level.is_set l' || is_local l' then
                equalize_variables false l' Level.set local
              else univ_inconsistency Le l r
            in
            Level.Set.fold fold levels local
          else
            univ_inconsistency Le l r
        end
      | ULevel r' ->
        (* We insert the constraint in the graph even if the graph
            already contains it.  Indeed, checking the existence of the
            constraint is costly when the constraint does not already
            exist directly as a single edge in the graph, but adding an
            edge in the graph which is implied by others is cheap.
            Hence, by doing this, we avoid a costly check here, and
            make further checks of this constraint easier since it will
            exist directly in the graph. *)
        match classify l with
        | USmall UProp ->
          { local with local_above_prop = Level.Set.add r' local.local_above_prop }
        | USmall USProp ->
          if UGraph.type_in_type univs then local
          else univ_inconsistency Le l r
        | USmall USet ->
          add_univ_local (Level.set, Le, r') local
        | ULevel l' ->
          add_univ_local (l', Le, r') local
        | UAlgebraic l ->
          enforce_leq_up l r' local
        | UMax (_, l) ->
          Univ.Level.Set.fold (fun l' accu -> add_univ_local (l', Le, r') accu) l local
      end
    | ULub (l, r) ->
      equalize_variables true l r local
    | UWeak (l, r) ->
      if weak_constraints ()
      then { local with local_weak = UPairSet.add (l, r) local.local_weak }
      else local
    | UEq (l, r) ->
      let local = unify_quality CONV l r local in
      let l = normalize_sort local.local_sorts l in
      let r = normalize_sort local.local_sorts r in
      equalize_universes l r local
  in
  let unify_universes cst local =
    try unify_universes cst local
    with
      | SortInconsistency e
        when QGraph.ignore_constraints (QState.elims local.local_sorts) -> local
      | SortInconsistency e as exn ->
        let info = Exninfo.info exn in
        Exninfo.iraise (UGraph.UniverseInconsistency e, info)
      | UGraph.UniverseInconsistency _ when UGraph.type_in_type univs -> local
  in
  let local = {
    local_cst = PConstraints.empty;
    local_weak = uctx.minim_extra.UnivMinim.weak_constraints;
    local_above_prop = uctx.minim_extra.UnivMinim.above_prop;
    local_sorts = uctx.sort_variables;
  } in
  let local = UnivProblem.Set.fold unify_universes cstrs local in
  let extra = { UnivMinim.above_prop = local.local_above_prop; UnivMinim.weak_constraints = local.local_weak } in
  let () = warn_template uctx (PConstraints.univs local.local_cst) in
  !vars, extra, local.local_cst, local.local_sorts

let add_constraints ?src uctx cstrs =
  let univs, local = uctx.local in
  let vars, extra, local', sorts = process_constraints uctx cstrs in
  { uctx with
    local = (univs, PConstraints.union local local');
    univ_variables = vars;
    universes = merge_univ_constraints uctx (PConstraints.univs local') uctx.universes;
    sort_variables =
          QState.merge_constraints (merge_elim_constraints ?src uctx (PConstraints.qualities local')) sorts ;
    minim_extra = extra; }

let set_above_prop q above uctx =
  match above with
  | AboveProp ->
    add_constraints uctx (UnivProblem.Set.singleton (QLeq (QConstant QProp, QVar q)))
  | BelowType ->
    match QState.set_above_prop q BelowType uctx.sort_variables with
    | Some sort_variables -> { uctx with sort_variables }
    | None ->
      (* TODO handle BelowType in unify_quality and use
         add_constraints here for more uniform erroring? *)
      CErrors.user_err Pp.(str "Could not enforce " ++ pr_uctx_qvar uctx q ++ str " <= Type.")

let problem_of_univ_constraints cstrs =
  UnivConstraints.fold (fun (l,d,r) acc ->
      let l = Universe.make l and r = sort_of_univ @@ Universe.make r in
      let cstr' = let open UnivProblem in
        match d with
        | Lt ->
          ULe (sort_of_univ @@ Universe.super l, r)
        | Le -> ULe (sort_of_univ l, r)
        | Eq -> UEq (sort_of_univ l, r)
      in UnivProblem.Set.add cstr' acc)
    cstrs UnivProblem.Set.empty

let problem_of_elim_constraints cstrs =
  ElimConstraints.fold (fun (l, k, r) pbs ->
      let open ElimConstraint in
      match k with
      | ElimTo -> UnivProblem.Set.add (QElimTo (l, r)) pbs)
    cstrs UnivProblem.Set.empty

let add_univ_constraints uctx cstrs =
  let cstrs = problem_of_univ_constraints cstrs in
  add_constraints ~src:Static uctx cstrs

let add_poly_constraints ?src uctx (qcstrs, ucstrs) =
  let lvl_pbs = problem_of_univ_constraints ucstrs in
  let elim_pbs = problem_of_elim_constraints qcstrs in
  let uctx = add_constraints ?src uctx (UnivProblem.Set.union lvl_pbs elim_pbs) in
  let local = on_snd (fun cst -> PConstraints.union cst (PConstraints.of_qualities qcstrs)) uctx.local in
  let sort_variables = QState.merge_constraints (fun cst -> merge_elim_constraints ?src uctx qcstrs cst) uctx.sort_variables in
  { uctx with local; sort_variables }

let check_elim_constraints uctx csts =
  Sorts.ElimConstraints.for_all (fun (l,k,r) ->
      let l = nf_quality uctx l in
      let r = nf_quality uctx r in
      match l,k,r with
        | _, ElimTo, _ -> Inductive.eliminates_to (QState.elims uctx.sort_variables) l r)
    csts

let check_eq_quality uctx q1 q2 =
  Sorts.Quality.equal q1 q2 || Sorts.Quality.equal (nf_quality uctx q1) (nf_quality uctx q2)

let check_constraint uctx (c:UnivProblem.t) =
  match c with
  | QEq (a,b) ->
    let a = nf_quality uctx a in
    let b = nf_quality uctx b in
    Quality.equal a b
  | QLeq (a,b) ->
    let a = nf_quality uctx a in
    let b = nf_quality uctx b in
    Quality.equal a b ||
      begin
        match a, b with
        | QConstant QProp, QConstant QType -> true
        | QConstant QProp, QVar q -> QState.is_above_prop uctx.sort_variables q
        | (QConstant _ | QVar _ | QGlobal _), _ -> false
      end
  | QElimTo (a, b) ->
    let a = nf_quality uctx a in
    let b = nf_quality uctx b in
    Inductive.eliminates_to (QState.elims uctx.sort_variables) a b
  | ULe (u,v) -> UGraph.check_leq_sort (fun q1 q2 -> check_eq_quality uctx q1 q2) uctx.universes u v
  | UEq (u,v) -> UGraph.check_eq_sort (fun q1 q2 -> check_eq_quality uctx q1 q2) uctx.universes u v
  | ULub (u,v) -> UGraph.check_eq_level uctx.universes u v
  | UWeak _ -> true

let check_constraints uctx csts =
  UnivProblem.Set.for_all (check_constraint uctx) csts

let constrain_variables diff uctx =
  let (us, (qcst, ucst)) = uctx.local in
  let (us, ucst), vars = UnivFlex.constrain_variables diff uctx.univ_variables (us, ucst) in
  { uctx with local = (us, (qcst, ucst)); univ_variables = vars }

type ('a, 'b, 'c, 'd) gen_universe_decl = {
  univdecl_qualities : 'a;
  univdecl_extensible_qualities : bool;
  univdecl_elim_constraints : 'b;
  univdecl_instance : 'c; (* Declared universes *)
  univdecl_extensible_instance : bool; (* Can new universes be added *)
  univdecl_univ_constraints : 'd; (* Declared univ constraints *)
  univdecl_extensible_constraints : bool; (* Can new constraints (elim or univ) be added *) }

type universe_decl =
  (QVar.t list, Sorts.ElimConstraints.t, Level.t list, Univ.UnivConstraints.t) gen_universe_decl

let default_univ_decl =
  { univdecl_qualities = [];
    (* in practice non named qualities will get collapsed for toplevel definitions,
       but side effects see named qualities from the surrounding definitions
       while using default_univ_decl *)
    univdecl_extensible_qualities = true;
    univdecl_elim_constraints = ElimConstraints.empty;
    univdecl_instance = [];
    univdecl_extensible_instance = true;
    univdecl_univ_constraints = UnivConstraints.empty;
    univdecl_extensible_constraints = true }

let univ_decl_csts decl =
  PConstraints.make decl.univdecl_elim_constraints decl.univdecl_univ_constraints

let pr_error_unbound_universes quals univs names =
  let open Pp in
  let nqs = QVar.Set.cardinal quals in
  let prqvar q =
    let info = QVar.Map.find_opt q (fst (snd names)) in
    h (pr_uctx_qvar_names names q ++ (match info with
        | None | Some {uloc=None} -> mt ()
        | Some {uloc=Some loc} -> spc() ++ str"(" ++ Loc.pr loc ++ str")"))
  in
  let nus = Level.Set.cardinal univs in
  let prlev u =
    let info = Level.Map.find_opt u (snd (snd names)) in
    h (pr_uctx_level_names names u ++ (match info with
        | None | Some {uloc=None} -> mt ()
        | Some {uloc=Some loc} -> spc() ++ str"(" ++ Loc.pr loc ++ str")"))
  in
  let ppqs = if nqs > 0 then
      str (if nqs = 1 then "Quality" else "Qualities") ++ spc () ++
      prlist_with_sep spc prqvar (QVar.Set.elements quals)
    else mt()
  in
  let ppus = if nus > 0 then
      let universe_s = CString.plural nus "universe" in
      let universe_s = if nqs = 0 then CString.capitalize_ascii universe_s else universe_s in
      str universe_s ++ spc () ++
      prlist_with_sep spc prlev (Level.Set.elements univs)
    else mt()
  in
  (hv 0
     (ppqs ++
      (if nqs > 0 && nus > 0 then strbrk " and " else mt()) ++
      ppus ++
      spc () ++ str (CString.conjugate_verb_to_be (nus + nqs)) ++ str" unbound."))

exception UnboundUnivs of QVar.Set.t * Level.Set.t * univ_names

(* XXX when we have multi location errors we won't have to pick an arbitrary error *)
let error_unbound_universes qs us uctx =
  let exception Found of Loc.t in
  let loc =
    try
      Level.Set.iter (fun u ->
          match Level.Map.find_opt u (snd (snd uctx)) with
          | None -> ()
          | Some info -> match info.uloc with
            | None -> ()
            | Some loc -> raise_notrace (Found loc))
        us;
      QVar.Set.iter (fun s ->
          match QVar.Map.find_opt s (fst (snd uctx)) with
          | None -> ()
          | Some info -> match info.uloc with
            | None -> ()
            | Some loc -> raise_notrace (Found loc))
        qs;
      None
    with Found loc -> Some loc
  in
  Loc.raise ?loc (UnboundUnivs (qs,us,uctx))

let () = CErrors.register_handler (function
    | UnboundUnivs (qs,us,uctx) -> Some (pr_error_unbound_universes qs us uctx)
    | _ -> None)

let universe_context_inst decl qvars levels names =
  let leftqs = List.fold_left (fun acc l -> QSet.remove l acc) qvars decl.univdecl_qualities in
  let leftus = List.fold_left (fun acc l -> Level.Set.remove l acc) levels decl.univdecl_instance in
  let () =
    let unboundqs = if decl.univdecl_extensible_qualities then QSet.empty else leftqs in
    let unboundus = if decl.univdecl_extensible_instance then Level.Set.empty else leftus in
    if not (QSet.is_empty unboundqs && Level.Set.is_empty unboundus)
    then error_unbound_universes unboundqs unboundus names
  in
  let leftqs = UContext.sort_qualities
      (Array.map_of_list (fun q -> Quality.QVar q) (QVar.Set.elements leftqs))
  in
  let leftus = UContext.sort_levels (Array.of_list (Level.Set.elements leftus)) in
  let instq = Array.append
      (Array.map_of_list (fun q -> QVar q) decl.univdecl_qualities)
      leftqs
  in
  let instu = Array.append (Array.of_list decl.univdecl_instance) leftus in
  let inst = Instance.of_array (instq,instu) in
  inst

let check_universe_context_set ~prefix levels names =
  let left =
    List.fold_left (fun left l -> Level.Set.remove l left)
      levels prefix
  in
  if not (Level.Set.is_empty left)
  then error_unbound_universes QVar.Set.empty left names

let check_univ_implication uctx cstrs cstrs' =
  let gr = uctx.initial_universes in
  let grext = merge_univ_constraints uctx cstrs gr in
  let cstrs' = UnivConstraints.filter (fun c -> not (UGraph.check_constraint grext c)) cstrs' in
  if UnivConstraints.is_empty cstrs' then ()
  else CErrors.user_err
      Pp.(str "Universe constraints are not implied by the ones declared: " ++
          UnivConstraints.pr (pr_uctx_level uctx) cstrs')

let check_elim_implication uctx cstrs cstrs' =
  let g = initial_elim_graph uctx in
  let grext = merge_elim_constraints ~src:Rigid uctx cstrs g in
  let cstrs' = ElimConstraints.filter (fun c -> not (QGraph.check_constraint grext c)) cstrs' in
  if ElimConstraints.is_empty cstrs' then ()
  else CErrors.user_err
      Pp.(str "Elimination constraints are not implied by the ones declared: " ++
          ElimConstraints.pr (quality_printer uctx) cstrs')

let check_implication uctx (elim_csts,univ_csts) (elim_csts',univ_csts') =
  check_univ_implication uctx univ_csts univ_csts';
  check_elim_implication uctx elim_csts elim_csts'

let check_template_univ_decl uctx ~template_qvars decl =
  let () =
    match List.filter (fun q -> not @@ QSet.mem q template_qvars) decl.univdecl_qualities with
    | (_ :: _) as qvars ->
      CErrors.user_err
        Pp.(str "Qualities " ++ prlist_with_sep spc (pr_uctx_qvar uctx) qvars ++
            str " cannot be template.")
    | [] ->
      if not (QVar.Set.equal template_qvars (QState.undefined uctx.sort_variables))
      then CErrors.anomaly Pp.(str "Bugged template univ declaration.")
  in
  (* XXX: when the kernel takes template entries closer to the polymorphic ones,
     we should perform some additional checks here. *)
  let () = assert (Sorts.ElimConstraints.is_empty decl.univdecl_elim_constraints) in
  let levels, csts = uctx.local in
  let () =
    let prefix = decl.univdecl_instance in
    if not decl.univdecl_extensible_instance
    then check_universe_context_set ~prefix levels uctx.names
  in
  if decl.univdecl_extensible_constraints then
    PContextSet.univ_context_set uctx.local
  else
    let () = check_implication uctx (univ_decl_csts decl) csts in
    (levels, decl.univdecl_univ_constraints)

let check_mono_univ_decl uctx decl =
  (* Note: if [decl] is [default_univ_decl], behave like [uctx.local] *)
  let () =
    if not (List.is_empty decl.univdecl_qualities)
    || not (QSet.is_empty (QState.undefined uctx.sort_variables))
    then CErrors.user_err Pp.(str "Monomorphic declarations may not have sort variables.")
  in
  let levels, csts = uctx.local in
  let () =
    let prefix = decl.univdecl_instance in
    if not decl.univdecl_extensible_instance
    then check_universe_context_set ~prefix levels uctx.names
  in
  if decl.univdecl_extensible_constraints then check_mono_sort_constraints uctx
  else
    let () = assert (Sorts.ElimConstraints.is_empty (fst csts)) in
    let () = check_implication uctx (univ_decl_csts decl) csts in
    levels, decl.univdecl_univ_constraints

let check_poly_univ_decl uctx decl =
  (* Note: if [decl] is [default_univ_decl], behave like [context uctx] *)
  let levels, (elim_csts,univ_csts) = uctx.local in
  let qvars = QState.undefined uctx.sort_variables in
  let inst = universe_context_inst decl qvars levels uctx.names in
  let nas = compute_instance_binders uctx inst in
  let univ_csts = if decl.univdecl_extensible_constraints then univ_csts
    else begin
      check_univ_implication uctx
        decl.univdecl_univ_constraints
        univ_csts;
      decl.univdecl_univ_constraints
    end
  in
  let elim_csts = if decl.univdecl_extensible_constraints then elim_csts
    else begin
      check_elim_implication uctx
        decl.univdecl_elim_constraints
        elim_csts;
      decl.univdecl_elim_constraints
    end
  in
  let uctx = UContext.make nas (inst, (elim_csts,univ_csts)) in
  uctx

let check_univ_decl ~poly uctx decl =
  let (binders, _) = uctx.names in
  let entry =
    if PolyFlags.univ_poly poly then Polymorphic_entry (check_poly_univ_decl uctx decl)
    else Monomorphic_entry (check_mono_univ_decl uctx decl)
  in
  entry, binders

let restrict_universe_context (univs, univ_csts) keep =
  let removed = Level.Set.diff univs keep in
  if Level.Set.is_empty removed then univs, univ_csts
  else
  let allunivs = UnivConstraints.fold (fun (u,_,v) all -> Level.Set.add u (Level.Set.add v all)) univ_csts univs in
  let g = UGraph.initial_universes in
  let g = Level.Set.fold (fun v g ->
      if Level.is_set v then g else
        UGraph.add_universe v ~strict:false g) allunivs g in
  let g = UGraph.merge_constraints univ_csts g in
  let allkept = Level.Set.union (UGraph.domain UGraph.initial_universes) (Level.Set.diff allunivs removed) in
  let univ_csts = UGraph.constraints_for ~kept:allkept g in
  let univ_csts = UnivConstraints.filter (fun (l,d,r) -> not (Level.is_set l && d == Le)) univ_csts in
  (Level.Set.inter univs keep, univ_csts)

let restrict_universe_pcontext (us, (qcst, ucst)) keep =
  let (us, ucst) = restrict_universe_context (us, ucst) keep in
  (us, (qcst, ucst))

let restrict uctx vars =
  let vars = Id.Map.fold (fun na l vars -> Level.Set.add l vars)
      (snd (fst uctx.names)) vars
  in
  let uctx' = restrict_universe_pcontext uctx.local vars in
  { uctx with local = uctx' }

let restrict_even_binders uctx vars =
  let uctx' = restrict_universe_pcontext uctx.local vars in
  { uctx with local = uctx' }

let restrict_univ_constraints uctx csts =
  let levels, (elim_csts,univ_csts) = uctx.local in
  let uctx' = { uctx with local = (levels,(elim_csts,UnivConstraints.empty)); universes = uctx.initial_universes } in
  add_univ_constraints uctx' csts

let restrict_elim_constraints ?src uctx csts =
  let levels, (elim_csts,univ_csts) = uctx.local in
  let g = initial_elim_graph uctx in
  (* XXX we are wreaking havoc with elimination constraints *)
  let sort_variables = QState.set_elims g uctx.sort_variables in
  let sort_variables = QState.merge_constraints (fun cst -> merge_elim_constraints ?src uctx elim_csts cst) sort_variables in
  { uctx with local = (levels, (csts, univ_csts)); sort_variables }

type rigid =
  | UnivRigid
  | UnivFlexible of bool (** Is substitution by an algebraic ok? *)

let univ_rigid = UnivRigid
let univ_flexible = UnivFlexible false
let univ_flexible_alg = UnivFlexible true

(** ~sideff indicates that it is ok to redeclare a universe.
    Also merges the universe context in the local constraint structures
    and not only in the graph. *)
let merge_universe_context_set ?loc ~sideff rigid uctx (levels, ucst) =
  let declare g =
    Level.Set.fold (fun u g ->
        try UGraph.add_universe ~strict:false u g
        with UGraph.AlreadyDeclared when sideff -> g)
      levels g
  in
  let names =
    let fold u accu =
      let update = function
        | None -> Some { uname = None; uloc = loc }
        | Some info -> match info.uloc with
          | None -> Some { info with uloc = loc }
          | Some _ -> Some info
      in
      Level.Map.update u update accu
    in
    (fst uctx.names, (fst (snd uctx.names), Level.Set.fold fold levels (snd (snd uctx.names))))
  in
  let initial = declare uctx.initial_universes in
  let univs = declare uctx.universes in
  let universes = merge_univ_constraints uctx ucst univs in
  let uctx =
    match rigid with
    | UnivRigid -> uctx
    | UnivFlexible b ->
      assert (not sideff);
      let uvars' = UnivFlex.add_levels levels ~algebraic:b uctx.univ_variables in
      { uctx with univ_variables = uvars' }
  in
  let (us, (qcst, ucst0)) = uctx.local in
  let local = (Univ.Level.Set.union us levels, (qcst, Univ.UnivConstraints.union ucst0 ucst)) in
  { uctx with names; local; universes;
              initial_universes = initial }

let merge_sort_variables ?loc ?(sort_rigid=false) ?src ~sideff uctx (qvars, csts) =
  let sort_variables =
    QVar.Set.fold (fun qv qstate -> QState.add ~check_fresh:(not sideff) ~rigid:sort_rigid qv qstate)
      qvars
      uctx.sort_variables
  in
  let names =
    let fold u accu =
      let update = function
        | None -> Some { uname = None; uloc = loc }
        | Some info -> match info.uloc with
          | None -> Some { info with uloc = loc }
          | Some _ -> Some info
      in
      QVar.Map.update u update accu
    in
    let qrev = QVar.Set.fold fold qvars (fst (snd uctx.names)) in
    (fst uctx.names, (qrev, snd (snd uctx.names)))
  in
  let sort_variables = QState.merge_constraints (merge_elim_constraints ?src uctx csts) sort_variables in
  let (us, (qcst, ucst)) = uctx.local in
  let local = (us, (Sorts.ElimConstraints.union qcst csts, ucst)) in
  { uctx with local; sort_variables; names }

let merge_sort_context_set ?loc ?sort_rigid ?src ~sideff rigid uctx ((qvars, levels), (qcst, ucst)) =
  let uctx = merge_sort_variables ?loc ?sort_rigid ?src ~sideff uctx (qvars, qcst) in
  merge_universe_context_set ?loc ~sideff rigid uctx (levels, ucst)

let demote_global_univs (lvl_set, univ_csts) uctx =
  let (local_univs, local_constraints) = uctx.local in
  let local_univs = Level.Set.diff local_univs lvl_set in
  let univ_variables = Level.Set.fold UnivFlex.remove lvl_set uctx.univ_variables in
  let update_ugraph g =
    let g = Level.Set.fold (fun u g ->
        try UGraph.add_universe u ~strict:true g
        with UGraph.AlreadyDeclared -> g)
        lvl_set
        g
    in
    UGraph.merge_constraints univ_csts g
  in
  let initial_universes = update_ugraph uctx.initial_universes in
  let universes = update_ugraph uctx.universes in
  { uctx with local = (local_univs, local_constraints); univ_variables; universes; initial_universes }

let demote_global_univ_entry entry uctx = match entry with
| Monomorphic_entry ucst ->
  demote_global_univs ucst uctx
| Polymorphic_entry _ -> uctx

(* Check bug_4363 bug_6323 bug_3539 and success/rewrite lemma l1
   for quick feedback when changing this code *)
let emit_side_effects eff u =
  let uctx = Safe_typing.universes_of_private eff in
  demote_global_univs uctx u

let merge_seff uctx uctx' =
  let levels = PContextSet.levels uctx' in
  let declare g =
    Level.Set.fold (fun u g ->
        try UGraph.add_universe ~strict:false u g
        with UGraph.AlreadyDeclared -> g)
      levels g
  in
  let initial_universes = declare uctx.initial_universes in
  let univs = declare uctx.universes in
  let universes = merge_univ_constraints uctx (PContextSet.univ_constraints uctx') univs in
  { uctx with universes; initial_universes }

let update_sigma_univs uctx univs =
  let eunivs =
    { uctx with
      initial_universes = univs;
      universes = univs }
  in
  merge_seff eunivs eunivs.local

let add_qnames ?loc s l ((qnames,unames), (qnames_rev,unames_rev)) =
  if Id.Map.mem s qnames
  then user_err ?loc
      Pp.(str "Quality " ++ Id.print s ++ str" already bound.");
  ((Id.Map.add s l qnames, unames),
   (QVar.Map.add l { uname = Some s; uloc = loc } qnames_rev, unames_rev))

let add_names ?loc s l ((qnames,unames), (qnames_rev,unames_rev)) =
  if UNameMap.mem s unames
  then user_err ?loc
      Pp.(str "Universe " ++ Id.print s ++ str" already bound.");
  ((qnames,UNameMap.add s l unames),
   (qnames_rev, Level.Map.add l { uname = Some s; uloc = loc } unames_rev))

let add_qloc l loc (names, (qnames_rev,unames_rev) as orig) =
  match loc with
  | None -> orig
  | Some _ -> (names, (QVar.Map.add l { uname = None; uloc = loc } qnames_rev, unames_rev))

let add_loc l loc (names, (qnames_rev,unames_rev) as orig) =
  match loc with
  | None -> orig
  | Some _ -> (names, (qnames_rev, Level.Map.add l { uname = None; uloc = loc } unames_rev))

let add_quality_variable ?loc ?(check_fresh=true) ~name ~rigid uctx q =
  let sort_variables = QState.add ~check_fresh ~rigid q uctx.sort_variables in
  let names = match name with
    | Some n -> add_qnames ?loc n q uctx.names
    | None -> add_qloc q loc uctx.names
  in
  { uctx with sort_variables; names }

let add_universe ?loc name strict uctx u =
  let initial_universes = UGraph.add_universe ~strict u uctx.initial_universes in
  let universes = UGraph.add_universe ~strict u uctx.universes in
  let local = PContextSet.add_level u uctx.local in
  let names =
    match name with
    | Some n -> add_names ?loc n u uctx.names
    | None -> add_loc u loc uctx.names
  in
  { uctx with names; local; initial_universes; universes }

let new_quality_variable ?loc ?(sort_rigid = false) ?name uctx =
  let q = UnivGen.fresh_sort_quality () in
  (* don't need to check_fresh as it's guaranteed new *)
  add_quality_variable ?loc ~name ~rigid:(sort_rigid || Option.has_some name) uctx q, q

let new_univ_level_variable ?loc rigid name uctx =
  let u = UnivGen.fresh_level () in
  let uctx =
    match rigid with
    | UnivRigid -> uctx
    | UnivFlexible algebraic ->
      let univ_variables = UnivFlex.add u ~algebraic uctx.univ_variables in
      { uctx with univ_variables }
  in
  let uctx = add_universe ?loc name false uctx u in
  uctx, u

let add_forgotten_univ uctx u = add_universe None true uctx u

let from_env env =
  { empty with
    universes = Environ.universes env;
    initial_universes = Environ.universes env;
    sort_variables = QState.of_elims (Environ.qualities env);
  }

let from_auctx ?(of_template=false) env auctx =
  let ustate = from_env env in
  let names = AbstractContext.names auctx in
  let name_to_option = function Name id -> Some id | Anonymous -> None in
  (* Inlined call to [AbstractContext.repr] to know what qvars, levels and constraints to add *)
  let add_qvar i ustate name =
    let q = QVar.make_var i in
    let ustate = add_quality_variable ~rigid:(not of_template) ~name:(name_to_option name) ustate q in
    if not of_template then ustate
    else set_above_prop q BelowType ustate
  in
  let ustate = Array.fold_left_i add_qvar ustate names.quals in
  let ustate = Array.fold_left_i (fun i ustate name -> add_universe (name_to_option name) false ustate (Level.var i)) ustate names.univs in
  let ustate = add_poly_constraints ustate (AbstractContext.constraints auctx) in
  ustate

let make_nonalgebraic_variable uctx u =
  { uctx with univ_variables = UnivFlex.make_nonalgebraic_variable uctx.univ_variables u }

let make_flexible_nonalgebraic uctx =
  { uctx with univ_variables = UnivFlex.make_all_undefined_nonalgebraic uctx.univ_variables }

let subst_univs_context_with_def def usubst (uctx, (elim_csts,univ_csts)) =
  (Level.Set.diff uctx def, PConstraints.make elim_csts @@
                              UnivSubst.subst_univs_constraints usubst univ_csts)

let normalize_univ_variables uctx =
  let normalized_variables, def, subst =
    UnivFlex.normalize_univ_variables uctx.univ_variables
  in
  let uctx_local = subst_univs_context_with_def def subst uctx.local in
  let univs = UGraph.merge_constraints (snd (snd uctx_local)) uctx.initial_universes in
  { uctx with
    local = uctx_local;
    univ_variables = normalized_variables;
    universes = univs }

let normalize_quality_variables uctx =
  let (lvls, (elim_cstrs, lvl_cstrs)) = uctx.local in
  let elim_cstrs = QState.normalize_elim_constraints uctx.sort_variables elim_cstrs in
  { uctx with local = (lvls, (elim_cstrs, lvl_cstrs)) }

let normalize_variables uctx =
  let uctx = normalize_univ_variables uctx in
  normalize_quality_variables uctx

let fix_undefined_variables uctx =
  { uctx with univ_variables = UnivFlex.fix_undefined_variables uctx.univ_variables }

let collapse_sort_variables ?except ~only_above_prop uctx =
  let sorts = QState.collapse ?except ~only_above_prop uctx.sort_variables in
  normalize_quality_variables { uctx with sort_variables = sorts }

let minimize uctx =
  let open UnivMinim in
  let (us, (qcst, ucst)) = uctx.local in
  let (vars', (us', ucst')) =
    normalize_context_set uctx.universes (us, ucst) uctx.univ_variables
      uctx.minim_extra
  in
  if Univ.ContextSet.equal (us', ucst') (us, ucst) then uctx
  else
    let universes = UGraph.merge_constraints ucst' uctx.initial_universes in
      { names = uctx.names;
        local = (us', (qcst, ucst'));
        univ_variables = vars';
        sort_variables = uctx.sort_variables;
        universes = universes;
        initial_universes = uctx.initial_universes;
        minim_extra = UnivMinim.empty_extra; (* weak constraints are consumed *) }

let universe_context_inst_decl decl qvars levels names =
  let leftqs = List.fold_left (fun acc l -> QSet.remove l acc) qvars decl.univdecl_qualities in
  let leftus = List.fold_left (fun acc l -> Level.Set.remove l acc) levels decl.univdecl_instance in
  let () =
    let unboundqs = if decl.univdecl_extensible_qualities then QSet.empty else leftqs in
    let unboundus = if decl.univdecl_extensible_instance then Level.Set.empty else leftus in
    if not (QSet.is_empty unboundqs && Level.Set.is_empty unboundus)
    then error_unbound_universes unboundqs unboundus names
  in
  let instq = Array.map_of_list (fun q -> QVar q) decl.univdecl_qualities in
  let instu = Array.of_list decl.univdecl_instance in
  let inst = Instance.of_array (instq,instu) in
  inst

let check_univ_decl_rev uctx decl =
  let levels, (elim_csts,univ_csts as csts) = uctx.local in
  let qvars = QState.undefined uctx.sort_variables in
  let inst = universe_context_inst_decl decl qvars levels uctx.names in
  let nas = compute_instance_binders uctx inst in
  let () = check_implication uctx csts (univ_decl_csts decl)
  in
  let uctx = fix_undefined_variables uctx in
  let uctx, univ_csts =
    if decl.univdecl_extensible_constraints
    then uctx, univ_csts
    else restrict_univ_constraints uctx decl.univdecl_univ_constraints,
         univ_csts
  in
  let uctx, elim_csts =
    if decl.univdecl_extensible_constraints
    then uctx, elim_csts
    else restrict_elim_constraints ~src:Rigid uctx decl.univdecl_elim_constraints,
         elim_csts
  in
  let uctx' = UContext.make nas (inst, (elim_csts,univ_csts)) in
  uctx, uctx'

let check_uctx_impl ~fail uctx uctx' =
  let levels, (elim_csts,univ_csts) = uctx'.local in
  let qvars_diff =
    QVar.Set.diff
      (QState.undefined uctx'.sort_variables)
      (QState.undefined uctx.sort_variables)
  in
  let levels_diff = Level.Set.diff levels (fst uctx.local) in
  let () = if not @@ (QVar.Set.is_empty qvars_diff && Level.Set.is_empty levels_diff) then
    error_unbound_universes qvars_diff levels_diff uctx'.names
  in
  let () =
    let grext = ugraph uctx in
    let cstrs' = UnivConstraints.filter (fun c -> not (UGraph.check_constraint grext c)) univ_csts in
    if UnivConstraints.is_empty cstrs' then ()
    else fail (UnivConstraints.pr (pr_uctx_level uctx) cstrs')
  in
  let () =
    let grext = elim_graph uctx in
    let cstrs' = ElimConstraints.filter (fun c -> not (QGraph.check_constraint grext c)) elim_csts in
    if ElimConstraints.is_empty cstrs' then ()
    else fail (ElimConstraints.pr (quality_printer uctx) cstrs')
  in
  ()


(* XXX print above_prop too *)
let pr_weak prl {minim_extra={UnivMinim.weak_constraints=weak; above_prop}} =
  let open Pp in
  v 0 (
    prlist_with_sep cut (fun (u,v) -> h (prl u ++ str " ~ " ++ prl v)) (UPairSet.elements weak)
    ++ if UPairSet.is_empty weak || Level.Set.is_empty above_prop then mt() else cut () ++
    prlist_with_sep cut (fun u -> h (str "Prop <= " ++ prl u)) (Level.Set.elements above_prop))

let pr_sort_opt_subst uctx =
  let local_name q = try Some (get_uname (QVar.Map.find q (fst @@ snd uctx.names)))
    with Not_found -> None
  in
  QState.pr (quality_printer uctx)
    local_name
    uctx.sort_variables

let pr ctx =
  let open Pp in
  let prl = pr_uctx_level ctx in
  if is_empty ctx then mt ()
  else
    v 0
      (str"UNIVERSES:"++brk(0,1)++
       h (PContextSet.pr (sort_printer ctx) (context_set ctx)) ++ fnl () ++
       UnivFlex.pr prl (subst ctx) ++ fnl() ++
       str"SORTS:"++brk(0,1)++
       h (pr_sort_opt_subst ctx) ++ fnl() ++
       str "WEAK CONSTRAINTS:"++brk(0,1)++
       h (pr_weak prl ctx) ++ fnl ())

module Internal =
struct

let reboot env uctx =
  let uctx_global = from_env env in
  { uctx_global with univ_variables = uctx.univ_variables; sort_variables = uctx.sort_variables }

end