File: univ.ml

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(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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)         *)
(************************************************************************)

(* Created in Caml by Gérard Huet for CoC 4.8 [Dec 1988] *)
(* Functional code by Jean-Christophe Filliâtre for Coq V7.0 [1999] *)
(* Extension with algebraic universes by HH for Coq V7.0 [Sep 2001] *)
(* Additional support for sort-polymorphic inductive types by HH [Mar 2006] *)

(* Revisions by Bruno Barras, Hugo Herbelin, Pierre Letouzey *)

open Pp
open CErrors
open Util

(* Universes are stratified by a partial ordering $\le$.
   Let $\~{}$ be the associated equivalence. We also have a strict ordering
   $<$ between equivalence classes, and we maintain that $<$ is acyclic,
   and contained in $\le$ in the sense that $[U]<[V]$ implies $U\le V$.

   At every moment, we have a finite number of universes, and we
   maintain the ordering in the presence of assertions $U<V$ and $U\le V$.

   The equivalence $\~{}$ is represented by a tree structure, as in the
   union-find algorithm. The assertions $<$ and $\le$ are represented by
   adjacency lists *)

module RawLevel =
struct
  open Names
  type t =
    | Prop
    | Set
    | Level of int * DirPath.t
    | Var of int

  (* Hash-consing *)

  let equal x y =
    x == y ||
      match x, y with
      | Prop, Prop -> true
      | Set, Set -> true
      | Level (n,d), Level (n',d') ->
        Int.equal n n' && DirPath.equal d d'
      | Var n, Var n' -> Int.equal n n'
      | _ -> false

  let compare u v =
    match u, v with
    | Prop,Prop -> 0
    | Prop, _ -> -1
    | _, Prop -> 1
    | Set, Set -> 0
    | Set, _ -> -1
    | _, Set -> 1
    | Level (i1, dp1), Level (i2, dp2) ->
      if i1 < i2 then -1
      else if i1 > i2 then 1
      else DirPath.compare dp1 dp2
    | Level _, _ -> -1
    | _, Level _ -> 1
    | Var n, Var m -> Int.compare n m

  open Hashset.Combine

  let hash = function
    | Prop -> combinesmall 1 0
    | Set -> combinesmall 1 1
    | Var n -> combinesmall 2 n
    | Level (n, d) -> combinesmall 3 (combine n (Names.DirPath.hash d))
end

module Level = struct

  open Names

  type raw_level = RawLevel.t =
  | Prop
  | Set
  | Level of int * DirPath.t
  | Var of int

  (** Embed levels with their hash value *)
  type t = { 
    hash : int;
    data : RawLevel.t }

  let equal x y = 
    x == y || Int.equal x.hash y.hash && RawLevel.equal x.data y.data

  let hash x = x.hash

  let data x = x.data

  let make l = { hash = RawLevel.hash l; data = l }

  let set = make Set
  let prop = make Prop
  let var i = make (Var i)
		  
  let is_small x = 
    match data x with
    | Level _ -> false
    | _ -> true

  let is_prop x =
    match data x with
    | Prop -> true
    | _ -> false

  let is_set x =
    match data x with
    | Set -> true
    | _ -> false

  let compare u v =
    if u == v then 0
    else
      let c = Int.compare (hash u) (hash v) in
	if c == 0 then RawLevel.compare (data u) (data v)
	else c
	    
  let to_string x = 
    match data x with
    | Prop -> "Prop"
    | Set -> "Set"
    | Level (n,d) -> Names.DirPath.to_string d^"."^string_of_int n
    | Var i -> "Var("^string_of_int i^")"

  let pr u = str (to_string u)

  let make m n = make (Level (n, m))

end

(** Level sets and maps *)
module LMap = HMap.Make (Level)
module LSet = struct
  include LMap.Set

  let pr s =
    str"{" ++ prlist_with_sep spc Level.pr (elements s) ++ str"}"

end

type 'a universe_map = 'a LMap.t

type universe_level = Level.t

type universe_level_subst_fn = universe_level -> universe_level

(* An algebraic universe [universe] is either a universe variable
   [Level.t] or a formal universe known to be greater than some
   universe variables and strictly greater than some (other) universe
   variables

   Universes variables denote universes initially present in the term
   to type-check and non variable algebraic universes denote the
   universes inferred while type-checking: it is either the successor
   of a universe present in the initial term to type-check or the
   maximum of two algebraic universes
*)

module Universe =
struct
  (* Invariants: non empty, sorted and without duplicates *)

  module Expr = 
  struct
    type t = Level.t * int
	
    let make l = (l, 0)

    let prop = (Level.prop, 0)
    let set = (Level.set, 0)
    let type1 = (Level.set, 1)

    let is_prop = function
      | (l,0) -> Level.is_prop l
      | _ -> false

    let equal x y = x == y ||
      (let (u,n) = x and (v,n') = y in
	 Int.equal n n' && Level.equal u v)

    let successor (u,n) =
      if Level.is_prop u then type1
      else (u, n + 1)

    let addn k (u,n as x) = 
      if k = 0 then x 
      else if Level.is_prop u then
	(Level.set,n+k)
      else (u,n+k)
	
    let super (u,n as x) (v,n' as y) =
      let cmp = Level.compare u v in
	if Int.equal cmp 0 then 
	  if n < n' then Inl true
	  else Inl false
	else if is_prop x then Inl true
	else if is_prop y then Inl false
	else Inr cmp

    let to_string (v, n) =
      if Int.equal n 0 then Level.to_string v
      else Level.to_string v ^ "+" ^ string_of_int n

    let pr x = str(to_string x)

    let level = function
      | (v,0) -> Some v
      | _ -> None

    let map f (v, n as x) = 
      let v' = f v in 
	if v' == v then x
	else if Level.is_prop v' && n != 0 then
	  (Level.set, n)
	else (v', n)

  end

  type t = Expr.t list

  let tip u = [u]
  let cons u v = u :: v

  let equal x y = x == y || List.equal Expr.equal x y

  let make l = tip (Expr.make l)

  let pr l = match l with
    | [u] -> Expr.pr u
    | _ -> 
      str "max(" ++ hov 0
	(prlist_with_sep pr_comma Expr.pr l) ++
        str ")"

  let level l = match l with
    | [l] -> Expr.level l
    | _ -> None

  (* The lower predicative level of the hierarchy that contains (impredicative)
     Prop and singleton inductive types *)
  let type0m = tip Expr.prop

  (* The level of sets *)
  let type0 = tip Expr.set

  (* When typing [Prop] and [Set], there is no constraint on the level,
     hence the definition of [type1_univ], the type of [Prop] *)    
  let type1 = tip (Expr.successor Expr.set)

  let is_type0m x = equal type0m x
  let is_type0 x = equal type0 x

  (* Returns the formal universe that lies juste above the universe variable u.
     Used to type the sort u. *)
  let super l = 
    List.map (fun x -> Expr.successor x) l

  let addn n l =
    List.map (fun x -> Expr.addn n x) l

  let rec merge_univs l1 l2 =
    match l1, l2 with
    | [], _ -> l2
    | _, [] -> l1
    | h1 :: t1, h2 :: t2 ->
      (match Expr.super h1 h2 with
      | Inl true (* h1 < h2 *) -> merge_univs t1 l2
      | Inl false -> merge_univs l1 t2
      | Inr c -> 
        if c <= 0 (* h1 < h2 is name order *)
	then cons h1 (merge_univs t1 l2)
	else cons h2 (merge_univs l1 t2))

  let sort u =
    let rec aux a l = 
      match l with
      | b :: l' ->
        (match Expr.super a b with
	| Inl false -> aux a l'
	| Inl true -> l
	| Inr c ->
	  if c <= 0 then cons a l
	  else cons b (aux a l'))
      | [] -> cons a l
    in 
      List.fold_right (fun a acc -> aux a acc) u []
	
  (* Returns the formal universe that is greater than the universes u and v.
     Used to type the products. *)
  let sup x y = merge_univs x y

  let empty = []

  let exists = List.exists

  let for_all = List.for_all

  let smart_map = List.Smart.map

end

type universe = Universe.t

(* The level of predicative Set *)
let type0m_univ = Universe.type0m
let type0_univ = Universe.type0
let type1_univ = Universe.type1
let is_type0m_univ = Universe.is_type0m
let is_type0_univ = Universe.is_type0
let is_univ_variable l = Universe.level l != None
let pr_uni = Universe.pr

let sup = Universe.sup
let super = Universe.super

open Universe

(* Comparison on this type is pointer equality *)
type canonical_arc =
    { univ: Level.t;
      lt: Level.t list;
      le: Level.t list;
      rank : int;
      predicative : bool}

let terminal u = {univ=u; lt=[]; le=[]; rank=0; predicative=false}

module UMap :
sig
  type key = Level.t
  type +'a t
  val empty : 'a t
  val add : key -> 'a -> 'a t -> 'a t
  val find : key -> 'a t -> 'a
  val fold : (key -> 'a -> 'b -> 'b) -> 'a t -> 'b -> 'b
end = HMap.Make(Level)

(* A Level.t is either an alias for another one, or a canonical one,
   for which we know the universes that are above *)

type univ_entry =
    Canonical of canonical_arc
  | Equiv of Level.t

type universes = univ_entry UMap.t

let enter_equiv_arc u v g =
  UMap.add u (Equiv v) g

let enter_arc ca g =
  UMap.add ca.univ (Canonical ca) g

(* Every Level.t has a unique canonical arc representative *)

(* repr : universes -> Level.t -> canonical_arc *)
(* canonical representative : we follow the Equiv links *)

let repr g u =
  let rec repr_rec u =
    let a =
      try UMap.find u g
      with Not_found -> anomaly ~label:"Univ.repr"
	  (str"Universe " ++ Level.pr u ++ str" undefined.")
    in
    match a with
      | Equiv v -> repr_rec v
      | Canonical arc -> arc
  in
  repr_rec u

let get_set_arc g = repr g Level.set

exception AlreadyDeclared
	    
let add_universe vlev strict g =
  try
    let _arcv = UMap.find vlev g in
      raise AlreadyDeclared
  with Not_found -> 
    let v = terminal vlev in
    let arc =
      let arc = get_set_arc g in
	if strict then
	  { arc with lt=vlev::arc.lt}
	else 
	  { arc with le=vlev::arc.le}
    in
    let g = enter_arc arc g in
      enter_arc v g
		       
(* reprleq : canonical_arc -> canonical_arc list *)
(* All canonical arcv such that arcu<=arcv with arcv#arcu *)
let reprleq g arcu =
  let rec searchrec w = function
    | [] -> w
    | v :: vl ->
	let arcv = repr g v in
        if List.memq arcv w || arcu==arcv then
	  searchrec w vl
        else
	  searchrec (arcv :: w) vl
  in
  searchrec [] arcu.le


(* between : Level.t -> canonical_arc -> canonical_arc list *)
(* between u v = { w | u<=w<=v, w canonical }          *)
(* between is the most costly operation *)

let between g arcu arcv =
  (* good are all w | u <= w <= v  *)
  (* bad are all w | u <= w ~<= v *)
    (* find good and bad nodes in {w | u <= w} *)
    (* explore b u = (b or "u is good") *)
  let rec explore ((good, bad, b) as input) arcu =
    if List.memq arcu good then
      (good, bad, true) (* b or true *)
    else if List.memq arcu bad then
      input    (* (good, bad, b or false) *)
    else
      let leq = reprleq g arcu in
	(* is some universe >= u good ? *)
      let good, bad, b_leq =
	List.fold_left explore (good, bad, false) leq
      in
	if b_leq then
	  arcu::good, bad, true (* b or true *)
	else
	  good, arcu::bad, b    (* b or false *)
  in
  let good,_,_ = explore ([arcv],[],false) arcu in
    good

(* We assume  compare(u,v) = LE with v canonical (see compare below).
   In this case List.hd(between g u v) = repr u
   Otherwise, between g u v = []
 *)

type constraint_type = Lt | Le | Eq

let constraint_type_ord c1 c2 = match c1, c2 with
| Lt, Lt -> 0
| Lt, _ -> -1
| Le, Lt -> 1
| Le, Le -> 0
| Le, Eq -> -1
| Eq, Eq -> 0
| Eq, _ -> 1

(** [compare_neq] : is [arcv] in the transitive upward closure of [arcu] ?

  In [strict] mode, we fully distinguish between LE and LT, while in
  non-strict mode, we simply answer LE for both situations.

  If [arcv] is encountered in a LT part, we could directly answer
  without visiting unneeded parts of this transitive closure.
  In [strict] mode, if [arcv] is encountered in a LE part, we could only
  change the default answer (1st arg [c]) from NLE to LE, since a strict
  constraint may appear later. During the recursive traversal,
  [lt_done] and [le_done] are universes we have already visited,
  they do not contain [arcv]. The 4rd arg is [(lt_todo,le_todo)],
  two lists of universes not yet considered, known to be above [arcu],
  strictly or not.

  We use depth-first search, but the presence of [arcv] in [new_lt]
  is checked as soon as possible : this seems to be slightly faster
  on a test.
*)

type fast_order = FastEQ | FastLT | FastLE | FastNLE

let fast_compare_neq strict g arcu arcv =
  (* [c] characterizes whether arcv has already been related
     to arcu among the lt_done,le_done universe *)
  let rec cmp c lt_done le_done lt_todo le_todo = match lt_todo, le_todo with
  | [],[] -> c
  | arc::lt_todo, le_todo ->
    if List.memq arc lt_done then
      cmp c lt_done le_done lt_todo le_todo
    else
      let rec find lt_todo lt le = match le with
      | [] ->
        begin match lt with
        | [] -> cmp c (arc :: lt_done) le_done lt_todo le_todo
        | u :: lt ->
          let arc = repr g u in
          if arc == arcv then
            if strict then FastLT else FastLE
          else find (arc :: lt_todo) lt le
        end
      | u :: le ->
        let arc = repr g u in
        if arc == arcv then
          if strict then FastLT else FastLE
        else find (arc :: lt_todo) lt le
      in
      find lt_todo arc.lt arc.le
  | [], arc::le_todo ->
    if arc == arcv then
      (* No need to continue inspecting universes above arc:
	 if arcv is strictly above arc, then we would have a cycle.
         But we cannot answer LE yet, a stronger constraint may
	 come later from [le_todo]. *)
      if strict then cmp FastLE lt_done le_done [] le_todo else FastLE
    else
      if (List.memq arc lt_done) || (List.memq arc le_done) then
        cmp c lt_done le_done [] le_todo
      else
        let rec find lt_todo lt = match lt with
        | [] ->
          let fold accu u =
            let node = repr g u in
            node :: accu
          in
          let le_new = List.fold_left fold le_todo arc.le in
          cmp c lt_done (arc :: le_done) lt_todo le_new
        | u :: lt ->
          let arc = repr g u in
          if arc == arcv then
            if strict then FastLT else FastLE
          else find (arc :: lt_todo) lt
        in
        find [] arc.lt
  in
  cmp FastNLE [] [] [] [arcu]

let fast_compare g arcu arcv =
  if arcu == arcv then FastEQ else fast_compare_neq true g arcu arcv

let is_leq g arcu arcv =
  arcu == arcv ||
    (match fast_compare_neq false g arcu arcv with
    | FastNLE -> false
    | (FastEQ|FastLE|FastLT) -> true)
    
let is_lt g arcu arcv =
  if arcu == arcv then false
  else
    match fast_compare_neq true g arcu arcv with
    | FastLT -> true
    | (FastEQ|FastLE|FastNLE) -> false

(* Invariants : compare(u,v) = EQ <=> compare(v,u) = EQ
                compare(u,v) = LT or LE => compare(v,u) = NLE
                compare(u,v) = NLE => compare(v,u) = NLE or LE or LT

   Adding u>=v is consistent iff compare(v,u) # LT
    and then it is redundant iff compare(u,v) # NLE
   Adding u>v is consistent iff compare(v,u) = NLE
    and then it is redundant iff compare(u,v) = LT *)

(** * Universe checks [check_eq] and [check_leq], used in coqchk *)

(** First, checks on universe levels *)

let check_equal g u v =
  let arcu = repr g u in
  let arcv = repr g v in
  arcu == arcv

let check_eq_level g u v = u == v || check_equal g u v

let is_set_arc u = Level.is_set u.univ
let is_prop_arc u = Level.is_prop u.univ

let check_smaller g strict u v =
  let arcu = repr g u in
  let arcv = repr g v in
  if strict then
    is_lt g arcu arcv
  else
    is_prop_arc arcu 
    || (is_set_arc arcu && arcv.predicative) 
    || is_leq g arcu arcv

(** Then, checks on universes *)

type 'a check_function = universes -> 'a -> 'a -> bool

let check_equal_expr g x y =
  x == y || (let (u, n) = x and (v, m) = y in 
	       Int.equal n m && check_equal g u v)

let check_eq_univs g l1 l2 =
  let f x1 x2 = check_equal_expr g x1 x2 in
  let exists x1 l = List.exists (fun x2 -> f x1 x2) l in
    List.for_all (fun x1 -> exists x1 l2) l1
    && List.for_all (fun x2 -> exists x2 l1) l2

let check_eq g u v =
  Universe.equal u v || check_eq_univs g u v

let check_smaller_expr g (u,n) (v,m) =
  let diff = n - m in
    match diff with
    | 0 -> check_smaller g false u v
    | 1 -> check_smaller g true u v
    | x when x < 0 -> check_smaller g false u v
    | _ -> false

let exists_bigger g ul l =
  Universe.exists (fun ul' -> 
    check_smaller_expr g ul ul') l

let real_check_leq g u v =
  Universe.for_all (fun ul -> exists_bigger g ul v) u
    
let check_leq g u v =
  Universe.equal u v ||
    Universe.is_type0m u ||
    check_eq_univs g u v || real_check_leq g u v

(** Enforcing new constraints : [setlt], [setleq], [merge], [merge_disc] *)

(** To speed up tests of Set </<= i *)
let set_predicative g arcv = 
  enter_arc {arcv with predicative = true} g

(* setlt : Level.t -> Level.t -> reason -> unit *)
(* forces u > v *)
(* this is normally an update of u in g rather than a creation. *)
let setlt g arcu arcv =
  let arcu' = {arcu with lt=arcv.univ::arcu.lt} in
  let g = 
    if is_set_arc arcu then set_predicative g arcv
    else g
  in
    enter_arc arcu' g, arcu'

(* checks that non-redundant *)
let setlt_if (g,arcu) v =
  let arcv = repr g v in
  if is_lt g arcu arcv then g, arcu
  else setlt g arcu arcv

(* setleq : Level.t -> Level.t -> unit *)
(* forces u >= v *)
(* this is normally an update of u in g rather than a creation. *)
let setleq g arcu arcv =
  let arcu' = {arcu with le=arcv.univ::arcu.le} in
  let g = 
    if is_set_arc arcu' then
      set_predicative g arcv
    else g
  in
    enter_arc arcu' g, arcu'

(* checks that non-redundant *)
let setleq_if (g,arcu) v =
  let arcv = repr g v in
  if is_leq g arcu arcv then g, arcu
  else setleq g arcu arcv

(* merge : Level.t -> Level.t -> unit *)
(* we assume  compare(u,v) = LE *)
(* merge u v  forces u ~ v with repr u as canonical repr *)
let merge g arcu arcv =
  (* we find the arc with the biggest rank, and we redirect all others to it *)
  let arcu, g, v =
    let best_ranked (max_rank, old_max_rank, best_arc, rest) arc =
      if Level.is_small arc.univ || arc.rank >= max_rank
      then (arc.rank, max_rank, arc, best_arc::rest)
      else (max_rank, old_max_rank, best_arc, arc::rest)
    in
      match between g arcu arcv with
      | [] -> anomaly (str "Univ.between.")
      | arc::rest ->
        let (max_rank, old_max_rank, best_arc, rest) =
          List.fold_left best_ranked (arc.rank, min_int, arc, []) rest in
          if max_rank > old_max_rank then best_arc, g, rest
          else begin
              (* one redirected node also has max_rank *)
            let arcu = {best_arc with rank = max_rank + 1} in
	      arcu, enter_arc arcu g, rest
          end 
  in
  let redirect (g,w,w') arcv =
    let g' = enter_equiv_arc arcv.univ arcu.univ g in
    (g',List.unionq arcv.lt w,arcv.le@w')
  in
  let (g',w,w') = List.fold_left redirect (g,[],[]) v in
  let g_arcu = (g',arcu) in
  let g_arcu = List.fold_left setlt_if g_arcu w in
  let g_arcu = List.fold_left setleq_if g_arcu w' in
  fst g_arcu

(* merge_disc : Level.t -> Level.t -> unit *)
(* we assume  compare(u,v) = compare(v,u) = NLE *)
(* merge_disc u v  forces u ~ v with repr u as canonical repr *)
let merge_disc g arc1 arc2 =
  let arcu, arcv = if arc1.rank < arc2.rank then arc2, arc1 else arc1, arc2 in
  let arcu, g = 
    if not (Int.equal arc1.rank arc2.rank) then arcu, g
    else
      let arcu = {arcu with rank = succ arcu.rank} in 
      arcu, enter_arc arcu g
  in
  let g' = enter_equiv_arc arcv.univ arcu.univ g in
  let g_arcu = (g',arcu) in
  let g_arcu = List.fold_left setlt_if g_arcu arcv.lt in
  let g_arcu = List.fold_left setleq_if g_arcu arcv.le in
  fst g_arcu

(* Universe inconsistency: error raised when trying to enforce a relation
   that would create a cycle in the graph of universes. *)

type univ_inconsistency = constraint_type * universe * universe

exception UniverseInconsistency of univ_inconsistency

let error_inconsistency o u v =
  raise (UniverseInconsistency (o,make u,make v))

(* enforc_univ_eq : Level.t -> Level.t -> unit *)
(* enforc_univ_eq u v will force u=v if possible, will fail otherwise *)

let enforce_univ_eq u v g =
  let arcu = repr g u in
  let arcv = repr g v in
    match fast_compare g arcu arcv with
    | FastEQ -> g
    | FastLT -> error_inconsistency Eq v u
    | FastLE -> merge g arcu arcv
    | FastNLE ->
      (match fast_compare g arcv arcu with
      | FastLT -> error_inconsistency Eq u v
      | FastLE -> merge g arcv arcu
      | FastNLE -> merge_disc g arcu arcv
      | FastEQ -> anomaly (Pp.str "Univ.compare."))

(* enforce_univ_leq : Level.t -> Level.t -> unit *)
(* enforce_univ_leq u v will force u<=v if possible, will fail otherwise *)
let enforce_univ_leq u v g =
  let arcu = repr g u in
  let arcv = repr g v in
  if is_leq g arcu arcv then g
  else
    match fast_compare g arcv arcu with
    | FastLT -> error_inconsistency Le u v
    | FastLE  -> merge g arcv arcu
    | FastNLE -> fst (setleq g arcu arcv)
    | FastEQ -> anomaly (Pp.str "Univ.compare.")

(* enforce_univ_lt u v will force u<v if possible, will fail otherwise *)
let enforce_univ_lt u v g =
  let arcu = repr g u in
  let arcv = repr g v in
    match fast_compare g arcu arcv with
    | FastLT -> g
    | FastLE -> fst (setlt g arcu arcv)
    | FastEQ -> error_inconsistency Lt u v
    | FastNLE ->
      match fast_compare_neq false g arcv arcu with
	FastNLE -> fst (setlt g arcu arcv)
      | FastEQ -> anomaly (Pp.str "Univ.compare.")
      | FastLE | FastLT -> error_inconsistency Lt u v

(* Prop = Set is forbidden here. *)
let initial_universes =
  let g = enter_arc (terminal Level.set) UMap.empty in
  let g = enter_arc (terminal Level.prop) g in
    enforce_univ_lt Level.prop Level.set g

(* Constraints and sets of constraints. *)    

type univ_constraint = Level.t * constraint_type * Level.t

let enforce_constraint cst g =
  match cst with
    | (u,Lt,v) -> enforce_univ_lt u v g
    | (u,Le,v) -> enforce_univ_leq u v g
    | (u,Eq,v) -> enforce_univ_eq u v g

module UConstraintOrd =
struct
  type t = univ_constraint
  let compare (u,c,v) (u',c',v') =
    let i = constraint_type_ord c c' in
    if not (Int.equal i 0) then i
    else
      let i' = Level.compare u u' in
      if not (Int.equal i' 0) then i'
      else Level.compare v v'
end

let pr_constraint_type op = 
  let op_str = match op with
    | Lt -> " < "
    | Le -> " <= "
    | Eq -> " = "
  in str op_str

module Constraint = 
struct 
  module S = Set.Make(UConstraintOrd)
  include S

  let pr prl c =
    fold (fun (u1,op,u2) pp_std ->
      pp_std ++ prl u1 ++ pr_constraint_type op ++
	prl u2 ++ fnl () )  c (str "")
end

let empty_constraint = Constraint.empty
let merge_constraints c g =
  Constraint.fold enforce_constraint c g
		  
type constraints = Constraint.t

(** A value with universe constraints. *)
type 'a constrained = 'a * constraints

(** Constraint functions. *)

type 'a constraint_function = 'a -> 'a -> constraints -> constraints

let check_constraint g (l,d,r) =
  match d with
  | Eq -> check_equal g l r
  | Le -> check_smaller g false l r
  | Lt -> check_smaller g true l r

let check_constraints c g =
  Constraint.for_all (check_constraint g) c

(**********************************************************************)
(** Universe polymorphism                                             *)
(**********************************************************************)

(** A universe level substitution, note that no algebraic universes are
    involved *)

type universe_level_subst = universe_level universe_map

(** A full substitution might involve algebraic universes *)
type universe_subst = universe universe_map

module Instance : sig 
    type t = Level.t array

    val empty : t
    val is_empty : t -> bool
    val equal : t -> t -> bool
    val subst_fn : universe_level_subst_fn -> t -> t
    val subst : universe_level_subst -> t -> t
    val pr : t -> Pp.t
    val check_eq : t check_function
    val length : t -> int
    val append : t -> t -> t
    val of_array : Level.t array -> t
end = 
struct
  type t = Level.t array

  let empty = [||]

  let is_empty x = Int.equal (Array.length x) 0

  let subst_fn fn t = 
    let t' = CArray.Smart.map fn t in
      if t' == t then t else t'

  let subst s t =
    let t' = 
      CArray.Smart.map (fun x -> try LMap.find x s with Not_found -> x) t
    in if t' == t then t else t'

  let pr =
    prvect_with_sep spc Level.pr

  let equal t u = 
    t == u ||
      (Array.is_empty t && Array.is_empty u) ||
      (CArray.for_all2 Level.equal t u 
	 (* Necessary as universe instances might come from different modules and 
	    unmarshalling doesn't preserve sharing *))

  let check_eq g t1 t2 =
    t1 == t2 ||
      (Int.equal (Array.length t1) (Array.length t2) &&
	 let rec aux i =
	   (Int.equal i (Array.length t1)) || (check_eq_level g t1.(i) t2.(i) && aux (i + 1))
	 in aux 0)

  let length = Array.length

  let append = Array.append

  let of_array i = i

end

(** Substitute instance inst for ctx in csts *)

let subst_instance_level s l =
  match l.Level.data with
  | Level.Var n -> s.(n) 
  | _ -> l

let subst_instance_instance s i = 
  Array.Smart.map (fun l -> subst_instance_level s l) i

let subst_instance_universe s u =
  let f x = Universe.Expr.map (fun u -> subst_instance_level s u) x in
  let u' = Universe.smart_map f u in
    if u == u' then u
    else Universe.sort u'

let subst_instance_constraint s (u,d,v as c) =
  let u' = subst_instance_level s u in
  let v' = subst_instance_level s v in
    if u' == u && v' == v then c
    else (u',d,v')

let subst_instance_constraints s csts =
  Constraint.fold 
    (fun c csts -> Constraint.add (subst_instance_constraint s c) csts)
    csts Constraint.empty

type universe_instance = Instance.t

type 'a puniverses = 'a * Instance.t
(** A context of universe levels with universe constraints,
    representiong local universe variables and constraints *)

module UContext =
struct
  type t = Instance.t constrained

  (** Universe contexts (variables as a list) *)
  let empty = (Instance.empty, Constraint.empty)
  let make x = x
  let instance (univs, cst) = univs
  let constraints (univs, cst) = cst
  let size (univs, _) = Instance.length univs

  let is_empty (univs, cst) = Instance.is_empty univs && Constraint.is_empty cst
  let pr prl (univs, cst as ctx) =
    if is_empty ctx then mt() else
      h 0 (Instance.pr univs ++ str " |= ") ++ h 0 (v 0 (Constraint.pr prl cst))
end

type universe_context = UContext.t

module AUContext =
struct
  include UContext

  let repr (inst, cst) =
    (Array.mapi (fun i l -> Level.var i) inst, cst)

  let instantiate inst (u, cst) =
    assert (Array.length u = Array.length inst);
    subst_instance_constraints inst cst

end

type abstract_universe_context = AUContext.t

module Variance =
struct
  (** A universe position in the instance given to a cumulative
     inductive can be the following. Note there is no Contravariant
     case because [forall x : A, B <= forall x : A', B'] requires [A =
     A'] as opposed to [A' <= A]. *)
  type t = Irrelevant | Covariant | Invariant

  let check_subtype x y = match x, y with
  | (Irrelevant | Covariant | Invariant), Irrelevant -> true
  | Irrelevant, Covariant -> false
  | (Covariant | Invariant), Covariant -> true
  | (Irrelevant | Covariant), Invariant -> false
  | Invariant, Invariant -> true

  let leq_constraint csts variance u u' =
    match variance with
    | Irrelevant -> csts
    | Covariant -> Constraint.add (u, Le, u') csts
    | Invariant -> Constraint.add (u, Eq, u') csts

  let eq_constraint csts variance u u' =
    match variance with
    | Irrelevant -> csts
    | Covariant | Invariant -> Constraint.add (u, Eq, u') csts

  let leq_constraints variance u u' csts =
    let len = Array.length u in
    assert (len = Array.length u' && len = Array.length variance);
    Array.fold_left3 leq_constraint csts variance u u'

  let eq_constraints variance u u' csts =
    let len = Array.length u in
    assert (len = Array.length u' && len = Array.length variance);
    Array.fold_left3 eq_constraint csts variance u u'
end

module CumulativityInfo =
struct
  type t = universe_context * Variance.t array

  let univ_context (univcst, subtypcst) = univcst
  let variance (univs, variance) = variance

end

module ACumulativityInfo = CumulativityInfo
type abstract_cumulativity_info = ACumulativityInfo.t

module ContextSet =
struct
  type t = LSet.t constrained
  let empty = LSet.empty, Constraint.empty
  let constraints (_, cst) = cst
  let levels (ctx, _) = ctx
  let make ctx cst = (ctx, cst)
end
type universe_context_set = ContextSet.t

(** Instance subtyping *)

let check_subtype univs ctxT ctx =
  if AUContext.size ctx == AUContext.size ctx then
    let (inst, cst) = AUContext.repr ctx in
    let cstT = UContext.constraints (AUContext.repr ctxT) in
    let push accu v = add_universe v false accu in
    let univs = Array.fold_left push univs inst in
    let univs = merge_constraints cstT univs in
    check_constraints cst univs
  else false

(** Substitutions. *)

let is_empty_subst = LMap.is_empty
let empty_level_subst = LMap.empty
let is_empty_level_subst = LMap.is_empty

(** Substitution functions *)

(** With level to level substitutions. *)
let subst_univs_level_level subst l =
  try LMap.find l subst
  with Not_found -> l

let subst_univs_level_universe subst u =
  let f x = Universe.Expr.map (fun u -> subst_univs_level_level subst u) x in
  let u' = Universe.smart_map f u in
    if u == u' then u
    else Universe.sort u'

let make_abstract_instance (ctx, _) = 
  Array.mapi (fun i l -> Level.var i) ctx

(** With level to universe substitutions. *)
type universe_subst_fn = universe_level -> universe

let make_subst subst = fun l -> LMap.find l subst

let subst_univs_expr_opt fn (l,n) =
  Universe.addn n (fn l)

let subst_univs_universe fn ul =
  let subst, nosubst = 
    List.fold_right (fun u (subst,nosubst) -> 
      try let a' = subst_univs_expr_opt fn u in
	    (a' :: subst, nosubst)
      with Not_found -> (subst, u :: nosubst))
      ul ([], [])
  in 
    if CList.is_empty subst then ul
    else 
      let substs = 
	List.fold_left Universe.merge_univs Universe.empty subst
      in
	List.fold_left (fun acc u -> Universe.merge_univs acc (Universe.tip u))
	  substs nosubst

let merge_context strict ctx g =
  let g = Array.fold_left
            (fun g v -> add_universe v strict g)
	    g (UContext.instance ctx)
  in merge_constraints (UContext.constraints ctx) g

let merge_context_set strict ctx g =
  let g = LSet.fold
   (* Include and side effects still have double-declared universes *)
	    (fun v g -> try add_universe v strict g with AlreadyDeclared -> g)
	    (ContextSet.levels ctx) g
  in merge_constraints (ContextSet.constraints ctx) g

(** Pretty-printing *)

let pr_constraints prl = Constraint.pr prl
    
let pr_universe_context = UContext.pr

let pr_arc = function
  | _, Canonical {univ=u; lt=[]; le=[]} ->
      mt ()
  | _, Canonical {univ=u; lt=lt; le=le} ->
      let opt_sep = match lt, le with
      | [], _ | _, [] -> mt ()
      | _ -> spc ()
      in
      Level.pr u ++ str " " ++
      v 0
        (pr_sequence (fun v -> str "< " ++ Level.pr v) lt ++
	 opt_sep ++
         pr_sequence (fun v -> str "<= " ++ Level.pr v) le) ++
      fnl ()
  | u, Equiv v ->
      Level.pr u  ++ str " = " ++ Level.pr v ++ fnl ()

let pr_universes g =
  let graph = UMap.fold (fun u a l -> (u,a)::l) g [] in
  prlist pr_arc graph