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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* 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] *)
(* Support for universe polymorphism by MS [2014] *)
(* Revisions by Bruno Barras, Hugo Herbelin, Pierre Letouzey, Matthieu
Sozeau, Pierre-Marie Pédrot *)
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
let hequal x y =
x == y ||
match x, y with
| Prop, Prop -> true
| Set, Set -> true
| Level (n,d), Level (n',d') ->
n == n' && d == d'
| Var n, Var n' -> n == n'
| _ -> false
let hcons = function
| Prop as x -> x
| Set as x -> x
| Level (n,d) as x ->
let d' = Names.DirPath.hcons d in
if d' == d then x else Level (n,d')
| Var n as x -> x
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
(** Hashcons on levels + their hash *)
module Self = struct
type nonrec t = t
type u = unit
let eq x y = x.hash == y.hash && RawLevel.hequal x.data y.data
let hash x = x.hash
let hashcons () x =
let data' = RawLevel.hcons x.data in
if x.data == data' then x else { x with data = data' }
end
let hcons =
let module H = Hashcons.Make(Self) in
Hashcons.simple_hcons H.generate H.hcons ()
let make l = hcons { hash = RawLevel.hash l; data = l }
let set = make Set
let prop = make Prop
let is_small x =
match data x with
| Level _ -> false
| Var _ -> false
| Prop -> true
| Set -> 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 natural_compare u v =
if u == v then 0
else RawLevel.compare (data u) (data v)
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 n -> "Var(" ^ string_of_int n ^ ")"
let pr u = str (to_string u)
let apart u v =
match data u, data v with
| Prop, Set | Set, Prop -> true
| _ -> false
let vars = Array.init 20 (fun i -> make (Var i))
let var n =
if n < 20 then vars.(n) else make (Var n)
let var_index u =
match data u with
| Var n -> Some n | _ -> None
let make m n = make (Level (n, Names.DirPath.hcons m))
end
(** Level maps *)
module LMap = struct
module M = HMap.Make (Level)
include M
let union l r =
merge (fun k l r ->
match l, r with
| Some _, _ -> l
| _, _ -> r) l r
let subst_union l r =
merge (fun k l r ->
match l, r with
| Some (Some _), _ -> l
| Some None, None -> l
| _, _ -> r) l r
let diff ext orig =
fold (fun u v acc ->
if mem u orig then acc
else add u v acc)
ext empty
let pr f m =
h 0 (prlist_with_sep fnl (fun (u, v) ->
Level.pr u ++ f v) (bindings m))
end
module LSet = struct
include LMap.Set
let pr prl s =
str"{" ++ prlist_with_sep spc prl (elements s) ++ str"}"
let of_array l =
Array.fold_left (fun acc x -> add x acc) empty l
end
type 'a universe_map = 'a LMap.t
type universe_level = Level.t
type universe_level_subst_fn = universe_level -> universe_level
type universe_set = LSet.t
(* 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
(* Hashing of expressions *)
module ExprHash =
struct
type t = Level.t * int
type u = Level.t -> Level.t
let hashcons hdir (b,n as x) =
let b' = hdir b in
if b' == b then x else (b',n)
let eq l1 l2 =
l1 == l2 ||
match l1,l2 with
| (b,n), (b',n') -> b == b' && n == n'
let hash (x, n) = n + Level.hash x
end
module H = Hashcons.Make(ExprHash)
let hcons =
Hashcons.simple_hcons H.generate H.hcons Level.hcons
let make l = (l, 0)
let compare u v =
if u == v then 0
else
let (x, n) = u and (x', n') = v in
if Int.equal n n' then Level.compare x x'
else n - n'
let prop = hcons (Level.prop, 0)
let set = hcons (Level.set, 0)
let type1 = hcons (Level.set, 1)
let is_small = function
| (l,0) -> Level.is_small 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 hash = ExprHash.hash
let leq (u,n) (v,n') =
let cmp = Level.compare u v in
if Int.equal cmp 0 then n <= n'
else if n <= n' then
(Level.is_prop u && Level.is_small v)
else false
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)
type super_result =
SuperSame of bool
(* The level expressions are in cumulativity relation. boolean
indicates if left is smaller than right? *)
| SuperDiff of int
(* The level expressions are unrelated, the comparison result
is canonical *)
(** [super u v] compares two level expressions,
returning [SuperSame] if they refer to the same level at potentially different
increments or [SuperDiff] if they are different. The booleans indicate if the
left expression is "smaller" than the right one in both cases. *)
let super (u,n as x) (v,n' as y) =
let cmp = Level.compare u v in
if Int.equal cmp 0 then SuperSame (n < n')
else
match x, y with
| (l,0), (l',0) ->
let open RawLevel in
(match Level.data l, Level.data l' with
| Prop, Prop -> SuperSame false
| Prop, _ -> SuperSame true
| _, Prop -> SuperSame false
| _, _ -> SuperDiff cmp)
| _, _ -> SuperDiff 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 pr_with f (v, n) =
if Int.equal n 0 then f v
else f v ++ str"+" ++ int n
let is_level = function
| (v, 0) -> true
| _ -> false
let level = function
| (v,0) -> Some v
| _ -> None
let get_level (v,n) = v
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 l = [l]
let cons x l = x :: l
let rec hash = function
| [] -> 0
| e :: l -> Hashset.Combine.combinesmall (Expr.ExprHash.hash e) (hash l)
let equal x y = x == y || List.equal Expr.equal x y
let compare x y = if x == y then 0 else List.compare Expr.compare x y
module Huniv = Hashcons.Hlist(Expr)
let hcons = Hashcons.recursive_hcons Huniv.generate Huniv.hcons Expr.hcons
let make l = tip (Expr.make l)
let tip x = tip x
let pr l = match l with
| [u] -> Expr.pr u
| _ ->
str "max(" ++ hov 0
(prlist_with_sep pr_comma Expr.pr l) ++
str ")"
let pr_with f l = match l with
| [u] -> Expr.pr_with f u
| _ ->
str "max(" ++ hov 0
(prlist_with_sep pr_comma (Expr.pr_with f) l) ++
str ")"
let is_level l = match l with
| [l] -> Expr.is_level l
| _ -> false
let rec is_levels l = match l with
| l :: r -> Expr.is_level l && is_levels r
| [] -> true
let level l = match l with
| [l] -> Expr.level l
| _ -> None
let levels l =
List.fold_left (fun acc x -> LSet.add (Expr.get_level x) acc) LSet.empty l
let is_small u =
match u with
| [l] -> Expr.is_small l
| _ -> false
(* 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 just above the universe variable u.
Used to type the sort u. *)
let super l =
if is_small l then type1
else
List.smartmap (fun x -> Expr.successor x) l
let addn n l =
List.smartmap (fun x -> Expr.addn n x) l
let rec merge_univs l1 l2 =
match l1, l2 with
| [], _ -> l2
| _, [] -> l1
| h1 :: t1, h2 :: t2 ->
let open Expr in
(match super h1 h2 with
| SuperSame true (* h1 < h2 *) -> merge_univs t1 l2
| SuperSame false -> merge_univs l1 t2
| SuperDiff 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' ->
let open Expr in
(match super a b with
| SuperSame false -> aux a l'
| SuperSame true -> l
| SuperDiff 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 smartmap = List.smartmap
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 is_small_univ = Universe.is_small
let pr_uni = Universe.pr
let sup = Universe.sup
let super = Universe.super
open Universe
let universe_level = Universe.level
type constraint_type = Lt | Le | Eq
type explanation = (constraint_type * universe) list
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
(* 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 * explanation option
exception UniverseInconsistency of univ_inconsistency
let error_inconsistency o u v (p:explanation option) =
raise (UniverseInconsistency (o,make u,make v,p))
(* Constraints and sets of constraints. *)
type univ_constraint = Level.t * constraint_type * Level.t
let pr_constraint_type op =
let op_str = match op with
| Lt -> " < "
| Le -> " <= "
| Eq -> " = "
in str op_str
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
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 "")
let universes_of c =
fold (fun (u1, op, u2) unvs -> LSet.add u2 (LSet.add u1 unvs)) c LSet.empty
end
let universes_of_constraints = Constraint.universes_of
let empty_constraint = Constraint.empty
let union_constraint = Constraint.union
let eq_constraint = Constraint.equal
type constraints = Constraint.t
module Hconstraint =
Hashcons.Make(
struct
type t = univ_constraint
type u = universe_level -> universe_level
let hashcons hul (l1,k,l2) = (hul l1, k, hul l2)
let eq (l1,k,l2) (l1',k',l2') =
l1 == l1' && k == k' && l2 == l2'
let hash = Hashtbl.hash
end)
module Hconstraints =
Hashcons.Make(
struct
type t = constraints
type u = univ_constraint -> univ_constraint
let hashcons huc s =
Constraint.fold (fun x -> Constraint.add (huc x)) s Constraint.empty
let eq s s' =
List.for_all2eq (==)
(Constraint.elements s)
(Constraint.elements s')
let hash = Hashtbl.hash
end)
let hcons_constraint = Hashcons.simple_hcons Hconstraint.generate Hconstraint.hcons Level.hcons
let hcons_constraints = Hashcons.simple_hcons Hconstraints.generate Hconstraints.hcons hcons_constraint
(** A value with universe constraints. *)
type 'a constrained = 'a * constraints
let constraints_of (_, cst) = cst
(** Constraint functions. *)
type 'a constraint_function = 'a -> 'a -> constraints -> constraints
let enforce_eq_level u v c =
(* We discard trivial constraints like u=u *)
if Level.equal u v then c
else if Level.apart u v then
error_inconsistency Eq u v None
else Constraint.add (u,Eq,v) c
let enforce_eq u v c =
match Universe.level u, Universe.level v with
| Some u, Some v -> enforce_eq_level u v c
| _ -> anomaly (Pp.str "A universe comparison can only happen between variables.")
let check_univ_eq u v = Universe.equal u v
let enforce_eq u v c =
if check_univ_eq u v then c
else enforce_eq u v c
let constraint_add_leq v u c =
(* We just discard trivial constraints like u<=u *)
if Expr.equal v u then c
else
match v, u with
| (x,n), (y,m) ->
let j = m - n in
if j = -1 (* n = m+1, v+1 <= u <-> v < u *) then
Constraint.add (x,Lt,y) c
else if j <= -1 (* n = m+k, v+k <= u <-> v+(k-1) < u *) then
if Level.equal x y then (* u+(k+1) <= u *)
raise (UniverseInconsistency (Le, Universe.tip v, Universe.tip u, None))
else anomaly (Pp.str"Unable to handle arbitrary u+k <= v constraints.")
else if j = 0 then
Constraint.add (x,Le,y) c
else (* j >= 1 *) (* m = n + k, u <= v+k *)
if Level.equal x y then c (* u <= u+k, trivial *)
else if Level.is_small x then c (* Prop,Set <= u+S k, trivial *)
else anomaly (Pp.str"Unable to handle arbitrary u <= v+k constraints.")
let check_univ_leq_one u v = Universe.exists (Expr.leq u) v
let check_univ_leq u v =
Universe.for_all (fun u -> check_univ_leq_one u v) u
let enforce_leq u v c =
let rec aux acc v =
match v with
| v :: l ->
aux (List.fold_right (fun u -> constraint_add_leq u v) u c) l
| [] -> acc
in aux c v
let enforce_leq u v c =
if check_univ_leq u v then c
else enforce_leq u v c
let enforce_leq_level u v c =
if Level.equal u v then c else Constraint.add (u,Le,v) c
let enforce_univ_constraint (u,d,v) =
match d with
| Eq -> enforce_eq u v
| Le -> enforce_leq u v
| Lt -> enforce_leq (super u) v
(* Miscellaneous functions to remove or test local univ assumed to
occur in a universe *)
let univ_level_mem u v =
List.exists (fun (l, n) -> Int.equal n 0 && Level.equal u l) v
let univ_level_rem u v min =
match Universe.level v with
| Some u' -> if Level.equal u u' then min else v
| None -> List.filter (fun (l, n) -> not (Int.equal n 0 && Level.equal u l)) v
(* Is u mentionned in v (or equals to v) ? *)
(**********************************************************************)
(** 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
let level_subst_of f =
fun l ->
try let u = f l in
match Universe.level u with
| None -> l
| Some l -> l
with Not_found -> l
module Instance : sig
type t = Level.t array
val empty : t
val is_empty : t -> bool
val of_array : Level.t array -> t
val to_array : t -> Level.t array
val append : t -> t -> t
val equal : t -> t -> bool
val length : t -> int
val hcons : t -> t
val hash : t -> int
val share : t -> t * int
val subst_fn : universe_level_subst_fn -> t -> t
val pr : (Level.t -> Pp.t) -> t -> Pp.t
val levels : t -> LSet.t
end =
struct
type t = Level.t array
let empty : t = [||]
module HInstancestruct =
struct
type nonrec t = t
type u = Level.t -> Level.t
let hashcons huniv a =
let len = Array.length a in
if Int.equal len 0 then empty
else begin
for i = 0 to len - 1 do
let x = Array.unsafe_get a i in
let x' = huniv x in
if x == x' then ()
else Array.unsafe_set a i x'
done;
a
end
let eq t1 t2 =
t1 == t2 ||
(Int.equal (Array.length t1) (Array.length t2) &&
let rec aux i =
(Int.equal i (Array.length t1)) || (t1.(i) == t2.(i) && aux (i + 1))
in aux 0)
let hash a =
let accu = ref 0 in
for i = 0 to Array.length a - 1 do
let l = Array.unsafe_get a i in
let h = Level.hash l in
accu := Hashset.Combine.combine !accu h;
done;
(* [h] must be positive. *)
let h = !accu land 0x3FFFFFFF in
h
end
module HInstance = Hashcons.Make(HInstancestruct)
let hcons = Hashcons.simple_hcons HInstance.generate HInstance.hcons Level.hcons
let hash = HInstancestruct.hash
let share a = (hcons a, hash a)
let empty = hcons [||]
let is_empty x = Int.equal (Array.length x) 0
let append x y =
if Array.length x = 0 then y
else if Array.length y = 0 then x
else Array.append x y
let of_array a =
assert(Array.for_all (fun x -> not (Level.is_prop x)) a);
a
let to_array a = a
let length a = Array.length a
let subst_fn fn t =
let t' = CArray.smartmap fn t in
if t' == t then t else of_array t'
let levels x = LSet.of_array x
let pr =
prvect_with_sep spc
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 *))
end
let enforce_eq_instances x y =
let ax = Instance.to_array x and ay = Instance.to_array y in
if Array.length ax != Array.length ay then
anomaly (Pp.(++) (Pp.str "Invalid argument: enforce_eq_instances called with")
(Pp.str " instances of different lengths."));
CArray.fold_right2 enforce_eq_level ax ay
let subst_instance_level s l =
match l.Level.data with
| Level.Var n -> s.(n)
| _ -> l
let subst_instance_instance s i =
Array.smartmap (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.smartmap 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
let out_punivs (x, y) = x
let in_punivs x = (x, Instance.empty)
let eq_puniverses f (x, u) (y, u') =
f x y && Instance.equal u u'
(** A context of universe levels with universe constraints,
representing local universe variables and constraints *)
module UContext =
struct
type t = Instance.t constrained
let make x = x
(** Universe contexts (variables as a list) *)
let empty = (Instance.empty, Constraint.empty)
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 prl univs ++ str " |= ") ++ h 0 (v 0 (Constraint.pr prl cst))
let hcons (univs, cst) =
(Instance.hcons univs, hcons_constraints cst)
let instance (univs, cst) = univs
let constraints (univs, cst) = cst
let union (univs, cst) (univs', cst') =
Instance.append univs univs', Constraint.union cst cst'
let dest x = x
let size (x,_) = Instance.length x
end
type universe_context = UContext.t
let hcons_universe_context = UContext.hcons
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
let hcons_abstract_universe_context = AUContext.hcons
(** Universe info for cumulative inductive types:
A context of universe levels
with universe constraints, representing local universe variables
and constraints, together with a context of universe levels with
universe constraints, representing conditions for subtyping used
for inductive types.
This data structure maintains the invariant that the context for
subtyping constraints is exactly twice as big as the context for
universe constraints. *)
module CumulativityInfo =
struct
type t = universe_context * universe_context
let make x =
if (Instance.length (UContext.instance (snd x))) =
(Instance.length (UContext.instance (fst x))) * 2 then x
else anomaly (Pp.str "Invalid subtyping information encountered!")
let empty = (UContext.empty, UContext.empty)
let is_empty (univcst, subtypcst) = UContext.is_empty univcst && UContext.is_empty subtypcst
let halve_context ctx =
let len = Array.length (Instance.to_array ctx) in
let halflen = len / 2 in
(Instance.of_array (Array.sub (Instance.to_array ctx) 0 halflen),
Instance.of_array (Array.sub (Instance.to_array ctx) halflen halflen))
let pr prl (univcst, subtypcst) =
if UContext.is_empty univcst then mt() else
let (ctx, ctx') = halve_context (UContext.instance subtypcst) in
(UContext.pr prl univcst) ++ fnl () ++ fnl () ++
h 0 (str "~@{" ++ Instance.pr prl ctx ++ str "} <= ~@{" ++ Instance.pr prl ctx' ++ str "} iff ")
++ fnl () ++ h 0 (v 0 (Constraint.pr prl (UContext.constraints subtypcst)))
let hcons (univcst, subtypcst) =
(UContext.hcons univcst, UContext.hcons subtypcst)
let univ_context (univcst, subtypcst) = univcst
let subtyp_context (univcst, subtypcst) = subtypcst
let create_trivial_subtyping ctx ctx' =
CArray.fold_left_i
(fun i cst l -> Constraint.add (l, Eq, Array.get ctx' i) cst)
Constraint.empty (Instance.to_array ctx)
(** This function takes a universe context representing constraints
of an inductive and a Instance.t of fresh universe names for the
subtyping (with the same length as the context in the given
universe context) and produces a UInfoInd.t that with the
trivial subtyping relation. *)
let from_universe_context univcst freshunivs =
let inst = (UContext.instance univcst) in
assert (Instance.length freshunivs = Instance.length inst);
(univcst, UContext.make (Instance.append inst freshunivs,
create_trivial_subtyping inst freshunivs))
let subtyping_susbst (univcst, subtypcst) =
let (ctx, ctx') = (halve_context (UContext.instance subtypcst))in
Array.fold_left2 (fun subst l1 l2 -> LMap.add l1 l2 subst) LMap.empty ctx ctx'
end
type cumulativity_info = CumulativityInfo.t
let hcons_cumulativity_info = CumulativityInfo.hcons
module ACumulativityInfo = CumulativityInfo
type abstract_cumulativity_info = ACumulativityInfo.t
let hcons_abstract_cumulativity_info = ACumulativityInfo.hcons
(** A set of universes with universe constraints.
We linearize the set to a list after typechecking.
Beware, representation could change.
*)
module ContextSet =
struct
type t = universe_set constrained
let empty = (LSet.empty, Constraint.empty)
let is_empty (univs, cst) = LSet.is_empty univs && Constraint.is_empty cst
let equal (univs, cst as x) (univs', cst' as y) =
x == y || (LSet.equal univs univs' && Constraint.equal cst cst')
let of_set s = (s, Constraint.empty)
let singleton l = of_set (LSet.singleton l)
let of_instance i = of_set (Instance.levels i)
let union (univs, cst as x) (univs', cst' as y) =
if x == y then x
else LSet.union univs univs', Constraint.union cst cst'
let append (univs, cst) (univs', cst') =
let univs = LSet.fold LSet.add univs univs' in
let cst = Constraint.fold Constraint.add cst cst' in
(univs, cst)
let diff (univs, cst) (univs', cst') =
LSet.diff univs univs', Constraint.diff cst cst'
let add_universe u (univs, cst) =
LSet.add u univs, cst
let add_constraints cst' (univs, cst) =
univs, Constraint.union cst cst'
let add_instance inst (univs, cst) =
let v = Instance.to_array inst in
let fold accu u = LSet.add u accu in
let univs = Array.fold_left fold univs v in
(univs, cst)
let sort_levels a =
Array.sort Level.natural_compare a; a
let to_context (ctx, cst) =
(Instance.of_array (sort_levels (Array.of_list (LSet.elements ctx))), cst)
let of_context (ctx, cst) =
(Instance.levels ctx, cst)
let pr prl (univs, cst as ctx) =
if is_empty ctx then mt() else
h 0 (LSet.pr prl univs ++ str " |= ") ++ h 0 (v 0 (Constraint.pr prl cst))
let constraints (univs, cst) = cst
let levels (univs, cst) = univs
end
type universe_context_set = ContextSet.t
(** A value in a universe context (resp. context set). *)
type 'a in_universe_context = 'a * universe_context
type 'a in_universe_context_set = 'a * universe_context_set
(** Substitutions. *)
let empty_subst = LMap.empty
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.smartmap f u in
if u == u' then u
else Universe.sort u'
let subst_univs_level_instance subst i =
let i' = Instance.subst_fn (subst_univs_level_level subst) i in
if i == i' then i
else i'
let subst_univs_level_constraint subst (u,d,v) =
let u' = subst_univs_level_level subst u
and v' = subst_univs_level_level subst v in
if d != Lt && Level.equal u' v' then None
else Some (u',d,v')
let subst_univs_level_constraints subst csts =
Constraint.fold
(fun c -> Option.fold_right Constraint.add (subst_univs_level_constraint subst c))
csts Constraint.empty
let subst_univs_level_abstract_universe_context subst (inst, csts) =
inst, subst_univs_level_constraints subst csts
(** 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 subst_univs_level fn l =
try Some (fn l)
with Not_found -> None
let subst_univs_constraint fn (u,d,v as c) cstrs =
let u' = subst_univs_level fn u in
let v' = subst_univs_level fn v in
match u', v' with
| None, None -> Constraint.add c cstrs
| Some u, None -> enforce_univ_constraint (u,d,make v) cstrs
| None, Some v -> enforce_univ_constraint (make u,d,v) cstrs
| Some u, Some v -> enforce_univ_constraint (u,d,v) cstrs
let subst_univs_constraints subst csts =
Constraint.fold
(fun c cstrs -> subst_univs_constraint subst c cstrs)
csts Constraint.empty
let make_instance_subst i =
let arr = Instance.to_array i in
Array.fold_left_i (fun i acc l ->
LMap.add l (Level.var i) acc)
LMap.empty arr
let make_inverse_instance_subst i =
let arr = Instance.to_array i in
Array.fold_left_i (fun i acc l ->
LMap.add (Level.var i) l acc)
LMap.empty arr
let make_abstract_instance (ctx, _) =
Array.mapi (fun i l -> Level.var i) ctx
let abstract_universes ctx =
let instance = UContext.instance ctx in
let subst = make_instance_subst instance in
let cstrs = subst_univs_level_constraints subst
(UContext.constraints ctx)
in
let ctx = UContext.make (instance, cstrs) in
subst, ctx
let abstract_cumulativity_info (univcst, substcst) =
let instance, univcst = abstract_universes univcst in
let _, substcst = abstract_universes substcst in
(instance, (univcst, substcst))
(** Pretty-printing *)
let pr_constraints prl = Constraint.pr prl
let pr_universe_context = UContext.pr
let pr_cumulativity_info = CumulativityInfo.pr
let pr_abstract_universe_context = AUContext.pr
let pr_abstract_cumulativity_info = ACumulativityInfo.pr
let pr_universe_context_set = ContextSet.pr
let pr_universe_subst =
LMap.pr (fun u -> str" := " ++ Universe.pr u ++ spc ())
let pr_universe_level_subst =
LMap.pr (fun u -> str" := " ++ Level.pr u ++ spc ())
module Huniverse_set =
Hashcons.Make(
struct
type t = universe_set
type u = universe_level -> universe_level
let hashcons huc s =
LSet.fold (fun x -> LSet.add (huc x)) s LSet.empty
let eq s s' =
LSet.equal s s'
let hash = Hashtbl.hash
end)
let hcons_universe_set =
Hashcons.simple_hcons Huniverse_set.generate Huniverse_set.hcons Level.hcons
let hcons_universe_context_set (v, c) =
(hcons_universe_set v, hcons_constraints c)
let hcons_univ x = Universe.hcons x
let explain_universe_inconsistency prl (o,u,v,p) =
let pr_uni = Universe.pr_with prl in
let pr_rel = function
| Eq -> str"=" | Lt -> str"<" | Le -> str"<="
in
let reason = match p with
| None | Some [] -> mt()
| Some p ->
str " because" ++ spc() ++ pr_uni v ++
prlist (fun (r,v) -> spc() ++ pr_rel r ++ str" " ++ pr_uni v)
p ++
(if Universe.equal (snd (List.last p)) u then mt() else
(spc() ++ str "= " ++ pr_uni u))
in
str "Cannot enforce" ++ spc() ++ pr_uni u ++ spc() ++
pr_rel o ++ spc() ++ pr_uni v ++ reason
let compare_levels = Level.compare
let eq_levels = Level.equal
let equal_universes = Universe.equal
|