From 001ff72b2c17fb7b2fcaefa2555c115f0d909a03 Mon Sep 17 00:00:00 2001
From: Matthieu Sozeau <mattam@mattam.org>
Date: Mon, 28 Oct 2013 14:08:46 +0100
Subject: Initial work on reintroducing old-style polymorphism for
 compatibility (the stdlib does not compile entirely).

---
 kernel/cooking.ml                              |  29 +++---
 kernel/declarations.mli                        |  23 ++++-
 kernel/declareops.ml                           | 130 ++++++++++---------------
 kernel/declareops.mli                          |   5 +
 kernel/environ.ml                              |  17 +++-
 kernel/environ.mli                             |   8 +-
 kernel/fast_typeops.ml                         |  33 ++++++-
 kernel/indtypes.ml                             |  22 +++--
 kernel/inductive.ml                            |  79 +++++++++++++--
 kernel/inductive.mli                           |   3 +
 kernel/subtyping.ml                            |   8 +-
 kernel/term_typing.ml                          |  38 ++++----
 kernel/typeops.ml                              |  56 +++++++++--
 kernel/typeops.mli                             |  20 ++--
 library/assumptions.ml                         |   2 +-
 library/declare.ml                             |   3 +-
 library/global.ml                              |  18 ++--
 library/impargs.ml                             |   2 +-
 library/universes.ml                           |  14 ++-
 plugins/Derive/g_derive.ml4                    |   2 +-
 plugins/extraction/extract_env.ml              |   2 +-
 plugins/extraction/extraction.ml               |  29 +++---
 plugins/extraction/table.ml                    |   2 +-
 plugins/funind/functional_principles_proofs.ml |   2 +-
 plugins/funind/indfun.ml                       |   2 +-
 plugins/funind/merge.ml                        |   2 +-
 plugins/xml/xmlcommand.ml                      |   2 +-
 pretyping/inductiveops.ml                      |  41 +++++++-
 pretyping/inductiveops.mli                     |   2 +-
 pretyping/retyping.ml                          |  10 +-
 pretyping/retyping.mli                         |   2 +-
 pretyping/typing.ml                            |   2 +-
 pretyping/vnorm.ml                             |   3 +-
 printing/prettyp.ml                            |   3 +-
 printing/printer.ml                            |   6 +-
 printing/printmod.ml                           |   3 +-
 proofs/logic.ml                                |   7 +-
 toplevel/discharge.ml                          |  27 +++--
 toplevel/search.ml                             |   7 +-
 toplevel/vernacentries.ml                      |   3 +-
 40 files changed, 445 insertions(+), 224 deletions(-)

diff --git a/kernel/cooking.ml b/kernel/cooking.ml
index 4bae6de66..3d580d713 100644
--- a/kernel/cooking.ml
+++ b/kernel/cooking.ml
@@ -183,21 +183,17 @@ let cook_constant env { from = cb; info = { Opaqueproof.modlist; abstract } } =
     Context.fold_named_context (fun (h,_,_) hyps ->
       List.filter (fun (id,_,_) -> not (Id.equal id h)) hyps)
       hyps ~init:cb.const_hyps in
-
-  (* let typ = match cb.const_type with *)
-  (*   | NonPolymorphicType t -> *)
-  (* 	let typ = *)
-  (*         abstract_constant_type (expmod_constr cache r.d_modlist t) hyps in *)
-  (* 	NonPolymorphicType typ *)
-  (*   | PolymorphicArity (ctx,s) -> *)
-  (* 	let t = mkArity (ctx,Type s.poly_level) in *)
-  (* 	let typ = *)
-  (*         abstract_constant_type (expmod_constr cache r.d_modlist t) hyps in *)
-  (* 	let j = make_judge (constr_of_def body) typ in *)
-  (* 	Typeops.make_polymorphic_if_constant_for_ind env j *)
-  (* in *)
-  let typ = 
-    abstract_constant_type (expmod_constr cache modlist cb.const_type) hyps 
+  let typ = match cb.const_type with
+    | RegularArity t ->
+  	let typ =
+          abstract_constant_type (expmod_constr cache modlist t) hyps in
+  	RegularArity typ
+    | TemplateArity (ctx,s) ->
+  	let t = mkArity (ctx,Type s.template_level) in
+  	let typ =
+          abstract_constant_type (expmod_constr cache modlist t) hyps in
+  	let j = make_judge (constr_of_def body) typ in
+  	Typeops.make_polymorphic_if_constant_for_ind env j
   in
   let projection pb =
     let c' = abstract_constant_body (expmod_constr cache modlist pb.proj_body) hyps in
@@ -210,6 +206,9 @@ let cook_constant env { from = cb; info = { Opaqueproof.modlist; abstract } } =
 	  | _ -> assert false 
       with Not_found -> (((pb.proj_ind,0),Univ.Instance.empty), 0)
     in 
+    let typ = (* By invariant, a regular arity *)
+      match typ with RegularArity t -> t | TemplateArity _ -> assert false 
+    in
     let ctx, ty' = decompose_prod_n (n' + pb.proj_npars + 1) typ in
       { proj_ind = mind; proj_npars = pb.proj_npars + n'; proj_arg = pb.proj_arg;
 	proj_type = ty'; proj_body = c' }
diff --git a/kernel/declarations.mli b/kernel/declarations.mli
index f3cb41f32..f269e165e 100644
--- a/kernel/declarations.mli
+++ b/kernel/declarations.mli
@@ -18,7 +18,24 @@ type engagement = ImpredicativeSet
 
 (** {6 Representation of constants (Definition/Axiom) } *)
 
-type constant_type = types
+(** Non-universe polymorphic mode polymorphism (Coq 8.2+): inductives
+    and constants hiding inductives are implicitely polymorphic when
+    applied to parameters, on the universes appearing in the whnf of
+    their parameters and their conclusion, in a template style.
+    
+    In truely universe polymorphic mode, we always use RegularArity.
+*)
+
+type template_arity = {
+  template_param_levels : Univ.universe option list;
+  template_level : Univ.universe;
+}
+
+type ('a, 'b) declaration_arity = 
+  | RegularArity of 'a
+  | TemplateArity of 'b
+
+type constant_type = (types, rel_context * template_arity) declaration_arity
 
 (** Inlining level of parameters at functor applications.
     None means no inlining *)
@@ -79,11 +96,13 @@ type wf_paths = recarg Rtree.t
 v}
 *)
 
-type inductive_arity = {
+type regular_inductive_arity = {
   mind_user_arity : types;
   mind_sort : sorts;
 }
 
+type inductive_arity = (regular_inductive_arity, template_arity) declaration_arity
+
 type one_inductive_body = {
 (** {8 Primitive datas } *)
 
diff --git a/kernel/declareops.ml b/kernel/declareops.ml
index 0e4b80495..be6c18810 100644
--- a/kernel/declareops.ml
+++ b/kernel/declareops.ml
@@ -13,6 +13,28 @@ open Util
 (** Operations concernings types in [Declarations] :
     [constant_body], [mutual_inductive_body], [module_body] ... *)
 
+(** {6 Arities } *)
+
+let subst_decl_arity f g sub ar = 
+  match ar with
+  | RegularArity x -> 
+    let x' = f sub x in 
+      if x' == x then ar
+      else RegularArity x'
+  | TemplateArity x -> 
+    let x' = g sub x in 
+      if x' == x then ar
+      else TemplateArity x'
+
+let map_decl_arity f g = function
+  | RegularArity a -> RegularArity (f a)
+  | TemplateArity a -> TemplateArity (g a)
+
+let hcons_template_arity ar =
+  { template_param_levels =
+      List.smartmap (Option.smartmap Univ.hcons_univ) ar.template_param_levels;
+    template_level = Univ.hcons_univ ar.template_level }
+
 (** {6 Constants } *)
 
 let body_of_constant cb = match cb.const_body with
@@ -44,14 +66,10 @@ let subst_rel_declaration sub (id,copt,t as x) =
 
 let subst_rel_context sub = List.smartmap (subst_rel_declaration sub)
 
-(* let subst_const_type sub arity = match arity with *)
-(*   | NonPolymorphicType s -> *)
-(*     let s' = subst_mps sub s in *)
-(*     if s==s' then arity else NonPolymorphicType s' *)
-(*   | PolymorphicArity (ctx,s) -> *)
-(*     let ctx' = subst_rel_context sub ctx in *)
-(*     if ctx==ctx' then arity else PolymorphicArity (ctx',s) *)
-
+let subst_template_cst_arity sub (ctx,s as arity) =
+  let ctx' = subst_rel_context sub ctx in
+    if ctx==ctx' then arity else (ctx',s)
+      
 let subst_const_type sub arity =
   if is_empty_subst sub then arity
   else subst_mps sub arity
@@ -73,7 +91,7 @@ let subst_const_body sub cb =
   if is_empty_subst sub then cb
   else
     let body' = subst_const_def sub cb.const_body in
-    let type' = subst_const_type sub cb.const_type in
+    let type' = subst_decl_arity subst_const_type subst_template_cst_arity sub cb.const_type in
     let proj' = Option.smartmap (subst_const_proj sub) cb.const_proj in
     if body' == cb.const_body && type' == cb.const_type
       && proj' == cb.const_proj then cb
@@ -102,7 +120,13 @@ let hcons_rel_decl ((n,oc,t) as d) =
 
 let hcons_rel_context l = List.smartmap hcons_rel_decl l
 
-let hcons_const_type t = Term.hcons_constr t
+let hcons_regular_const_arity t = Term.hcons_constr t
+
+let hcons_template_const_arity (ctx, ar) = 
+  (hcons_rel_context ctx, hcons_template_arity ar)
+
+let hcons_const_type = 
+  map_decl_arity hcons_regular_const_arity hcons_template_const_arity
 
 let hcons_const_def = function
   | Undef inl -> Undef inl
@@ -164,72 +188,15 @@ let subst_wf_paths sub p = Rtree.smartmap (subst_recarg sub) p
 
 (** {7 Substitution of inductive declarations } *)
 
-(* OLD POLYMORPHISM *)
-(* let subst_indarity sub ar = match ar with *)
-(*   | Monomorphic s -> *)
-(*     let uar' = subst_mps sub s.mind_user_arity in *)
-(*     if uar' == s.mind_user_arity then ar *)
-(*     else Monomorphic { mind_user_arity = uar'; mind_sort = s.mind_sort } *)
-(*   | Polymorphic _ -> ar *)
-
-(* let subst_mind_packet sub mip = *)
-(*   let { mind_nf_lc = nf; *)
-(*         mind_user_lc = user; *)
-(*         mind_arity_ctxt = ctxt; *)
-(*         mind_arity = ar; *)
-(*         mind_recargs = ra } = mip *)
-(*   in *)
-(*   let nf' = Array.smartmap (subst_mps sub) nf in *)
-(*   let user' = *)
-(*     (\* maintain sharing of [mind_user_lc] and [mind_nf_lc] *\) *)
-(*     if user==nf then nf' *)
-(*     else Array.smartmap (subst_mps sub) user *)
-(*   in *)
-(*   let ctxt' = subst_rel_context sub ctxt in *)
-(*   let ar' = subst_indarity sub ar in *)
-(*   let ra' = subst_wf_paths sub ra in *)
-(*   if nf==nf' && user==user' && ctxt==ctxt' && ar==ar' && ra==ra' *)
-(*   then mip *)
-(*   else *)
-(*     { mip with *)
-(*       mind_nf_lc = nf'; *)
-(*       mind_user_lc = user'; *)
-(*       mind_arity_ctxt = ctxt'; *)
-(*       mind_arity = ar'; *)
-(*       mind_recargs = ra' } *)
-
-(* let subst_mind sub mib = *)
-(*   assert (List.is_empty mib.mind_hyps); (\* we're outside sections *\) *)
-(*   if is_empty_subst sub then mib *)
-(*   else *)
-(*     let params = mib.mind_params_ctxt in *)
-(*     let params' = Context.map_rel_context (subst_mps sub) params in *)
-(*     let packets = mib.mind_packets in *)
-(*     let packets' = Array.smartmap (subst_mind_packet sub) packets in *)
-(*     if params==params' && packets==packets' then mib *)
-(*     else *)
-(*       { mib with *)
-(*         mind_params_ctxt = params'; *)
-(*         mind_packets = packets'; *)
-(*         mind_native_name = ref NotLinked } *)
-
-(* (\** {6 Hash-consing of inductive declarations } *\) *)
-
-(* (\** Just as for constants, this hash-consing is quite partial *\) *)
-
-(* let hcons_indarity = function *)
-(*   | Monomorphic a -> *)
-(*     Monomorphic *)
-(*       { mind_user_arity = Term.hcons_constr a.mind_user_arity; *)
-(*         mind_sort = Term.hcons_sorts a.mind_sort } *)
-(*   | Polymorphic a -> Polymorphic (hcons_polyarity a) *)
+let subst_regular_ind_arity sub s =
+  let uar' = subst_mps sub s.mind_user_arity in
+    if uar' == s.mind_user_arity then s 
+    else { mind_user_arity = uar'; mind_sort = s.mind_sort }
 
-(** Substitution of inductive declarations *)
+let subst_template_ind_arity sub s = s
 
-let subst_indarity sub s =
-  { mind_user_arity = subst_mps sub s.mind_user_arity;
-    mind_sort = s.mind_sort;
-  }
+let subst_ind_arity =
+  subst_decl_arity subst_regular_ind_arity subst_template_ind_arity
 
 let subst_mind_packet sub mbp =
   { mind_consnames = mbp.mind_consnames;
@@ -238,7 +205,7 @@ let subst_mind_packet sub mbp =
     mind_typename = mbp.mind_typename;
     mind_nf_lc = Array.smartmap (subst_mps sub) mbp.mind_nf_lc;
     mind_arity_ctxt = subst_rel_context sub mbp.mind_arity_ctxt;
-    mind_arity = subst_indarity sub mbp.mind_arity;
+    mind_arity = subst_ind_arity sub mbp.mind_arity;
     mind_user_lc = Array.smartmap (subst_mps sub) mbp.mind_user_lc;
     mind_nrealargs = mbp.mind_nrealargs;
     mind_nrealargs_ctxt = mbp.mind_nrealargs_ctxt;
@@ -261,12 +228,19 @@ let subst_mind_body sub mib =
     mind_polymorphic = mib.mind_polymorphic;
     mind_universes = mib.mind_universes }
 
-(** Hash-consing of inductive declarations *)
+(** {6 Hash-consing of inductive declarations } *)
 
-let hcons_indarity a =
+let hcons_regular_ind_arity a =
   { mind_user_arity = Term.hcons_constr a.mind_user_arity;
     mind_sort = Term.hcons_sorts a.mind_sort }
 
+(** Just as for constants, this hash-consing is quite partial *)
+
+let hcons_ind_arity =
+  map_decl_arity hcons_regular_ind_arity hcons_template_arity
+
+(** Substitution of inductive declarations *)
+
 let hcons_mind_packet oib =
   let user = Array.smartmap Term.hcons_types oib.mind_user_lc in
   let nf = Array.smartmap Term.hcons_types oib.mind_nf_lc in
@@ -275,7 +249,7 @@ let hcons_mind_packet oib =
   { oib with
     mind_typename = Names.Id.hcons oib.mind_typename;
     mind_arity_ctxt = hcons_rel_context oib.mind_arity_ctxt;
-    mind_arity = hcons_indarity oib.mind_arity;
+    mind_arity = hcons_ind_arity oib.mind_arity;
     mind_consnames = Array.smartmap Names.Id.hcons oib.mind_consnames;
     mind_user_lc = user;
     mind_nf_lc = nf }
diff --git a/kernel/declareops.mli b/kernel/declareops.mli
index 0c5f3584e..9f5197668 100644
--- a/kernel/declareops.mli
+++ b/kernel/declareops.mli
@@ -12,6 +12,11 @@ open Mod_subst
 (** Operations concerning types in [Declarations] :
     [constant_body], [mutual_inductive_body], [module_body] ... *)
 
+(** {6 Arities} *)
+
+val map_decl_arity : ('a -> 'c) -> ('b -> 'd) ->
+  ('a, 'b) declaration_arity -> ('c, 'd) declaration_arity
+
 (** {6 Constants} *)
 
 val subst_const_body : substitution -> constant_body -> constant_body
diff --git a/kernel/environ.ml b/kernel/environ.ml
index 323d6fcea..1f29fd67a 100644
--- a/kernel/environ.ml
+++ b/kernel/environ.ml
@@ -214,12 +214,22 @@ let universes_and_subst_of cb u =
   let subst = Univ.make_universe_subst u univs in
     subst, Univ.instantiate_univ_context subst univs
 
+let get_regular_arity = function
+  | RegularArity a -> a
+  | TemplateArity _ -> assert false
+
+let map_regular_arity f = function
+  | RegularArity a as ar -> 
+    let a' = f a in 
+      if a' == a then ar else RegularArity a'
+  | TemplateArity _ -> assert false
+
 (* constant_type gives the type of a constant *)
 let constant_type env (kn,u) =
   let cb = lookup_constant kn env in
     if cb.const_polymorphic then
       let subst, csts = universes_and_subst_of cb u in
-	(subst_univs_constr subst cb.const_type, csts)
+	(map_regular_arity (subst_univs_constr subst) cb.const_type, csts)
     else cb.const_type, Univ.Constraint.empty
 
 let constant_type_in_ctx env kn =
@@ -260,7 +270,8 @@ let constant_value_and_type env (kn, u) =
 	| Def l_body -> Some (subst_univs_constr subst (Mod_subst.force_constr l_body))
 	| OpaqueDef _ -> None
 	| Undef _ -> None
-      in b', subst_univs_constr subst cb.const_type, cst
+      in
+	b', map_regular_arity (subst_univs_constr subst) cb.const_type, cst
     else 
       let b' = match cb.const_body with
 	| Def l_body -> Some (Mod_subst.force_constr l_body)
@@ -277,7 +288,7 @@ let constant_type_in env (kn,u) =
   let cb = lookup_constant kn env in
     if cb.const_polymorphic then
       let subst = Univ.make_universe_subst u (Future.force cb.const_universes) in
-	subst_univs_constr subst cb.const_type
+	map_regular_arity (subst_univs_constr subst) cb.const_type
     else cb.const_type
 
 let constant_value_in env (kn,u) =
diff --git a/kernel/environ.mli b/kernel/environ.mli
index fb5d79718..330d9c277 100644
--- a/kernel/environ.mli
+++ b/kernel/environ.mli
@@ -140,12 +140,12 @@ type const_evaluation_result = NoBody | Opaque | IsProj
 exception NotEvaluableConst of const_evaluation_result
 
 val constant_value : env -> constant puniverses -> constr constrained
-val constant_type : env -> constant puniverses -> types constrained
-val constant_type_in_ctx : env -> constant -> types Univ.in_universe_context
+val constant_type : env -> constant puniverses -> constant_type constrained
+val constant_type_in_ctx : env -> constant -> constant_type Univ.in_universe_context
 
 val constant_opt_value : env -> constant puniverses -> (constr * Univ.constraints) option
 val constant_value_and_type : env -> constant puniverses -> 
-  types option * constr * Univ.constraints
+  constr option * constant_type * Univ.constraints
 (** The universe context associated to the constant, empty if not 
     polymorphic *)
 val constant_context : env -> constant -> Univ.universe_context
@@ -154,7 +154,7 @@ val constant_context : env -> constant -> Univ.universe_context
    already contains the constraints corresponding to the constant 
    application. *)
 val constant_value_in : env -> constant puniverses -> constr
-val constant_type_in : env -> constant puniverses -> types
+val constant_type_in : env -> constant puniverses -> constant_type
 val constant_opt_value_in : env -> constant puniverses -> constr option
 
 
diff --git a/kernel/fast_typeops.ml b/kernel/fast_typeops.ml
index b441e02a3..8b7230c3b 100644
--- a/kernel/fast_typeops.ml
+++ b/kernel/fast_typeops.ml
@@ -109,17 +109,40 @@ let check_hyps_inclusion env f c sign =
 
 (* Type of constants *)
 
-let type_of_constant env cst = constant_type env cst
-let type_of_constant_in env cst = constant_type_in env cst
-let type_of_constant_knowing_parameters env t _ = t
 
-let judge_of_constant env (kn,u as cst) =
+let type_of_constant_knowing_parameters_arity env t paramtyps =
+  match t with
+  | RegularArity t -> t
+  | TemplateArity (sign,ar) ->
+      let ctx = List.rev sign in
+      let ctx,s = instantiate_universes env ctx ar paramtyps in
+      mkArity (List.rev ctx,s)
+
+let type_of_constant_knowing_parameters env cst paramtyps =
+  let ty, cu = constant_type env cst in
+    type_of_constant_knowing_parameters_arity env ty paramtyps, cu
+
+let type_of_constant_type env t =
+  type_of_constant_knowing_parameters_arity env t [||]
+
+let type_of_constant env cst =
+  type_of_constant_knowing_parameters env cst [||]
+
+let type_of_constant_in env cst =
+  let ar = constant_type_in env cst in
+    type_of_constant_knowing_parameters_arity env ar [||]
+
+let judge_of_constant_knowing_parameters env (kn,u as cst) jl =
   let cb = lookup_constant kn env in
   let () = check_hyps_inclusion env mkConstU cst cb.const_hyps in
-  let ty, cu = type_of_constant env cst in
+  let paramstyp = Array.map (fun j -> lazy j.uj_type) jl in
+  let ty, cu = type_of_constant_knowing_parameters env cst paramstyp in
   let () = check_constraints cu env in
     ty
 
+let judge_of_constant env cst =
+  judge_of_constant_knowing_parameters env cst [||]
+
 let type_of_projection env (cst,u) =
   let cb = lookup_constant cst env in
   match cb.const_proj with
diff --git a/kernel/indtypes.ml b/kernel/indtypes.ml
index 0ac6a4e4a..e89bbf0d9 100644
--- a/kernel/indtypes.ml
+++ b/kernel/indtypes.ml
@@ -300,7 +300,7 @@ let typecheck_inductive env ctx mie =
 	     Universe.pr defu ++ Pp.str " declared for inductive type, inferred level is "
 	     ++ Universe.pr infu)
       in
-	(id,cn,lc,(sign,(not is_natural,full_arity,defu))))
+	(id,cn,lc,(sign,RegularArity (not is_natural,full_arity,defu))))
     inds
   in (env_arities, params, inds)
 
@@ -615,9 +615,13 @@ let allowed_sorts is_smashed s =
 (* - all_sorts in case of small, unitary Prop (not smashed) *)
 (* - logical_sorts in case of large, unitary Prop (smashed) *)
 
+let arity_conclusion = function
+  | RegularArity (_, c, _) -> c
+  | TemplateArity s -> mkType s.template_level
+
 let fold_inductive_blocks f =
   Array.fold_left (fun acc (_,_,lc,(arsign,ar)) ->
-    f (Array.fold_left f acc lc) (it_mkProd_or_LetIn (pi2 ar) arsign))
+    f (Array.fold_left f acc lc) (it_mkProd_or_LetIn (arity_conclusion ar) arsign))
 
 let used_section_variables env inds =
   let ids = fold_inductive_blocks
@@ -675,12 +679,14 @@ let build_inductive env p ctx env_ar params kn isrecord isfinite inds nmr recarg
 	splayed_lc in
     (* Elimination sorts *)
     let arkind,kelim = 
-      let (info,ar,defs) = ar_kind in
-      let s = sort_of_univ defs in
-      let kelim = allowed_sorts info s in
-	{ mind_user_arity = ar;
-	  mind_sort = s;
-	}, kelim in
+      match ar_kind with
+      | TemplateArity ar -> TemplateArity ar, all_sorts
+      | RegularArity (info,ar,defs) ->
+	let s = sort_of_univ defs in
+	let kelim = allowed_sorts info s in
+	let ar = RegularArity 
+	  { mind_user_arity = ar; mind_sort = s; } in
+	  ar, kelim in
     (* Assigning VM tags to constructors *)
     let nconst, nblock = ref 0, ref 0 in
     let transf num =
diff --git a/kernel/inductive.ml b/kernel/inductive.ml
index e57b0b4ad..9862ffd3e 100644
--- a/kernel/inductive.ml
+++ b/kernel/inductive.ml
@@ -152,30 +152,93 @@ let sort_as_univ = function
 | Prop Null -> Universe.type0m
 | Prop Pos -> Universe.type0
 
+(* Template polymorphism *)
+
 let cons_subst u su subst =
   try
     (u, Universe.sup su (List.assoc_f Universe.eq u subst)) ::
       List.remove_assoc_f Universe.eq u subst
   with Not_found -> (u, su) :: subst
 
+let actualize_decl_level env lev t =
+  let sign,s = dest_arity env t in
+  mkArity (sign,lev)
+
+(* Bind expected levels of parameters to actual levels *)
+(* Propagate the new levels in the signature *)
+let rec make_subst env = function
+  | (_,Some _,_ as t)::sign, exp, args ->
+      let ctx,subst = make_subst env (sign, exp, args) in
+      t::ctx, subst
+  | d::sign, None::exp, args ->
+      let args = match args with _::args -> args | [] -> [] in
+      let ctx,subst = make_subst env (sign, exp, args) in
+      d::ctx, subst
+  | d::sign, Some u::exp, a::args ->
+      (* We recover the level of the argument, but we don't change the *)
+      (* level in the corresponding type in the arity; this level in the *)
+      (* arity is a global level which, at typing time, will be enforce *)
+      (* to be greater than the level of the argument; this is probably *)
+      (* a useless extra constraint *)
+      let s = sort_as_univ (snd (dest_arity env (Lazy.force a))) in
+      let ctx,subst = make_subst env (sign, exp, args) in
+      d::ctx, cons_subst u s subst
+  | (na,None,t as d)::sign, Some u::exp, [] ->
+      (* No more argument here: we instantiate the type with a fresh level *)
+      (* which is first propagated to the corresponding premise in the arity *)
+      (* (actualize_decl_level), then to the conclusion of the arity (via *)
+      (* the substitution) *)
+      let ctx,subst = make_subst env (sign, exp, []) in
+	d::ctx, subst
+  | sign, [], _ ->
+      (* Uniform parameters are exhausted *)
+      sign,[]
+  | [], _, _ ->
+      assert false
+
 exception SingletonInductiveBecomesProp of Id.t
 
+let instantiate_universes env ctx ar argsorts =
+  let args = Array.to_list argsorts in
+  let ctx,subst = make_subst env (ctx,ar.template_param_levels,args) in
+  let level = subst_large_constraints subst ar.template_level in
+  let ty =
+    (* Singleton type not containing types are interpretable in Prop *)
+    if is_type0m_univ level then prop_sort
+    (* Non singleton type not containing types are interpretable in Set *)
+    else if is_type0_univ level then set_sort
+    (* This is a Type with constraints *)
+    else Type level
+  in
+    (ctx, ty)
+
 (* Type of an inductive type *)
 
-let type_of_inductive_gen env ((mib,mip),u) =
-  let subst = make_inductive_subst mib u in
-    (subst_univs_constr subst mip.mind_arity.mind_user_arity, subst)
+let type_of_inductive_gen ?(polyprop=true) env ((mib,mip),u) paramtyps =
+  match mip.mind_arity with
+  | RegularArity a ->
+    let subst = make_inductive_subst mib u in
+      (subst_univs_constr subst a.mind_user_arity, subst)
+  | TemplateArity ar ->
+    let ctx = List.rev mip.mind_arity_ctxt in
+    let ctx,s = instantiate_universes env ctx ar paramtyps in
+      (* The Ocaml extraction cannot handle (yet?) "Prop-polymorphism", i.e.
+         the situation where a non-Prop singleton inductive becomes Prop
+         when applied to Prop params *)
+      if not polyprop && not (is_type0m_univ ar.template_level) && is_prop_sort s
+      then raise (SingletonInductiveBecomesProp mip.mind_typename);
+      mkArity (List.rev ctx,s), Univ.LMap.empty
 
 let type_of_inductive env pind = 
-  fst (type_of_inductive_gen env pind)
+  fst (type_of_inductive_gen env pind [||])
 
 let constrained_type_of_inductive env ((mib,mip),u as pind) =
-  let ty, subst = type_of_inductive_gen env pind in
+  let ty, subst = type_of_inductive_gen env pind [||] in
   let cst = instantiate_inductive_constraints mib subst in
     (ty, cst)
 
 let type_of_inductive_knowing_parameters env ?(polyprop=false) mip args = 
-  type_of_inductive env mip
+  fst (type_of_inductive_gen env mip args)
 
 (* The max of an array of universes *)
 
@@ -248,7 +311,9 @@ let local_rels ctxt =
 (* Get type of inductive, with parameters instantiated *)
 
 let inductive_sort_family mip =
-  family_of_sort mip.mind_arity.mind_sort
+  match mip.mind_arity with
+  | RegularArity s -> family_of_sort s.mind_sort
+  | TemplateArity _ -> InType
 
 let mind_arity mip =
   mip.mind_arity_ctxt, inductive_sort_family mip
diff --git a/kernel/inductive.mli b/kernel/inductive.mli
index c4a7452f0..a403003e2 100644
--- a/kernel/inductive.mli
+++ b/kernel/inductive.mli
@@ -109,6 +109,9 @@ exception SingletonInductiveBecomesProp of Id.t
 
 val max_inductive_sort : sorts array -> universe
 
+val instantiate_universes : env -> rel_context ->
+  template_arity -> constr Lazy.t array -> rel_context * sorts
+
 (** {6 Debug} *)
 
 type size = Large | Strict
diff --git a/kernel/subtyping.ml b/kernel/subtyping.ml
index 2c093939a..56e94ba0d 100644
--- a/kernel/subtyping.ml
+++ b/kernel/subtyping.ml
@@ -277,8 +277,8 @@ let check_constant cst env mp1 l info1 cb2 spec2 subst1 subst2 =
       let cb1 = Declareops.subst_const_body subst1 cb1 in
       let cb2 = Declareops.subst_const_body subst2 cb2 in
       (* Start by checking types*)
-      let typ1 = cb1.const_type in
-      let typ2 = cb2.const_type in
+      let typ1 = Typeops.type_of_constant_type env cb1.const_type in
+      let typ2 = Typeops.type_of_constant_type env cb2.const_type in
       let cst = check_type cst env typ1 typ2 in
       (* Now we check the bodies:
 	 - A transparent constant can only be implemented by a compatible
@@ -309,7 +309,7 @@ let check_constant cst env mp1 l info1 cb2 spec2 subst1 subst2 =
       let arity1,cst1 = constrained_type_of_inductive env 
 	((mind1,mind1.mind_packets.(i)),u1) in
       let cst2 = UContext.constraints (Future.force cb2.const_universes) in
-      let typ2 = cb2.const_type in
+      let typ2 = Typeops.type_of_constant_type env cb2.const_type in
       let cst = Constraint.union cst (Constraint.union cst1 cst2) in
       let error = NotConvertibleTypeField (env, arity1, typ2) in
        check_conv error cst infer_conv_leq env arity1 typ2
@@ -324,7 +324,7 @@ let check_constant cst env mp1 l info1 cb2 spec2 subst1 subst2 =
       let u1 = inductive_instance mind1 in
       let ty1,cst1 = constrained_type_of_constructor (cstr,u1) (mind1,mind1.mind_packets.(i)) in
       let cst2 = UContext.constraints (Future.force cb2.const_universes) in
-      let ty2 = cb2.const_type in
+      let ty2 = Typeops.type_of_constant_type env cb2.const_type in
       let cst = Constraint.union cst (Constraint.union cst1 cst2) in
       let error = NotConvertibleTypeField (env, ty1, ty2) in
        check_conv error cst infer_conv env ty1 ty2
diff --git a/kernel/term_typing.ml b/kernel/term_typing.ml
index a7f2125a7..c655e2f33 100644
--- a/kernel/term_typing.ml
+++ b/kernel/term_typing.ml
@@ -29,17 +29,20 @@ let prerr_endline =
   if debug then prerr_endline else fun _ -> ()
 
 let constrain_type env j poly = function
-  | `None -> j.uj_type
+  | `None -> 
+    if not poly then (* Old-style polymorphism *)
+      make_polymorphic_if_constant_for_ind env j
+    else RegularArity j.uj_type
   | `Some t ->
       let tj = infer_type env t in
       let _ = judge_of_cast env j DEFAULTcast tj in
 	assert (eq_constr t tj.utj_val);
-	t
+	RegularArity t
   | `SomeWJ (t, tj) ->
       let tj = infer_type env t in
       let _ = judge_of_cast env j DEFAULTcast tj in
 	assert (eq_constr t tj.utj_val);
-	t
+	RegularArity t
 
 let map_option_typ = function None -> `None | Some x -> `Some x
 
@@ -72,12 +75,12 @@ let handle_side_effects env body side_eff =
       | Undef _ -> assert false
       | Def b ->
           let b = Mod_subst.force_constr b in
-          let b_ty = cb.const_type in
+          let b_ty = Typeops.type_of_constant_type env cb.const_type in
           let t = sub c 1 (Vars.lift 1 t) in
           mkLetIn (cname c, b, b_ty, t)
       | OpaqueDef b -> 
           let b = Opaqueproof.force_proof b in
-          let b_ty = cb.const_type in
+          let b_ty = Typeops.type_of_constant_type env cb.const_type in
           let t = sub c 1 (Vars.lift 1 t) in
           mkApp (mkLambda (cname c, b_ty, t), [|b|]) in
     List.fold_right fix_body cbl t
@@ -98,7 +101,7 @@ let infer_declaration env kn dcl =
       let env = push_context uctx env in
       let j = infer env t in
       let t = hcons_constr (Typeops.assumption_of_judgment env j) in
-      Undef nl, t, None, poly, Future.from_val uctx, false, ctx
+      Undef nl, RegularArity t, None, poly, Future.from_val uctx, false, ctx
   | DefinitionEntry ({ const_entry_type = Some typ;
                        const_entry_opaque = true } as c) ->
       let env = push_context (Future.join c.const_entry_universes) env in (* FIXME *)
@@ -113,7 +116,7 @@ let infer_declaration env kn dcl =
           feedback_completion_typecheck feedback_id;
           j.uj_val, Univ.empty_constraint) in
       let def = OpaqueDef (Opaqueproof.create proofterm) in
-      def, typ, None, c.const_entry_polymorphic, c.const_entry_universes,
+      def, RegularArity typ, None, c.const_entry_polymorphic, c.const_entry_universes,
 	c.const_entry_inline_code, c.const_entry_secctx
   | DefinitionEntry c ->
       let env = push_context (Future.join c.const_entry_universes) env in (* FIXME *)
@@ -129,13 +132,13 @@ let infer_declaration env kn dcl =
       def, typ, None, c.const_entry_polymorphic,
         c.const_entry_universes, c.const_entry_inline_code, c.const_entry_secctx
 
-(* let global_vars_set_constant_type env = function *)
-(*   | NonPolymorphicType t -> global_vars_set env t *)
-(*   | PolymorphicArity (ctx,_) -> *)
-(*       Context.fold_rel_context *)
-(*         (fold_rel_declaration *)
-(* 	  (fun t c -> Id.Set.union (global_vars_set env t) c)) *)
-(*       ctx ~init:Id.Set.empty *)
+let global_vars_set_constant_type env = function
+  | RegularArity t -> global_vars_set env t
+  | TemplateArity (ctx,_) ->
+      Context.fold_rel_context
+        (fold_rel_declaration
+	  (fun t c -> Id.Set.union (global_vars_set env t) c))
+      ctx ~init:Id.Set.empty
 
 let record_aux env s1 s2 =
   let v =
@@ -162,7 +165,7 @@ let build_constant_declaration kn env (def,typ,proj,poly,univs,inline_code,ctx)
     | None when not (List.is_empty context_ids) -> 
         (* No declared section vars, and non-empty section context:
            we must look at the body NOW, if any *)
-        let ids_typ = global_vars_set env typ in
+        let ids_typ = global_vars_set_constant_type env typ in
         let ids_def = match def with
         | Undef _ -> Idset.empty
         | Def cs -> global_vars_set env (Mod_subst.force_constr cs)
@@ -186,14 +189,14 @@ let build_constant_declaration kn env (def,typ,proj,poly,univs,inline_code,ctx)
         match def with
         | Undef _ as x -> x (* nothing to check *)
         | Def cs as x ->
-            let ids_typ = global_vars_set env typ in
+            let ids_typ = global_vars_set_constant_type env typ in
             let ids_def = global_vars_set env (Mod_subst.force_constr cs) in
             let inferred = keep_hyps env (Idset.union ids_typ ids_def) in
             check declared inferred;
             x
         | OpaqueDef lc -> (* In this case we can postpone the check *)
             OpaqueDef (Opaqueproof.iter_direct_opaque (fun c ->
-              let ids_typ = global_vars_set env typ in
+              let ids_typ = global_vars_set_constant_type env typ in
               let ids_def = global_vars_set env c in
               let inferred = keep_hyps env (Idset.union ids_typ ids_def) in
               check declared inferred) lc) in
@@ -229,6 +232,7 @@ let translate_recipe env kn r =
 let translate_local_def env id centry =
   let def,typ,proj,poly,univs,inline_code,ctx =
     infer_declaration env None (DefinitionEntry centry) in
+  let typ = type_of_constant_type env typ in
   def, typ, univs
 
 (* Insertion of inductive types. *)
diff --git a/kernel/typeops.ml b/kernel/typeops.ml
index 09fd4cc7f..09e2fb1d5 100644
--- a/kernel/typeops.ml
+++ b/kernel/typeops.ml
@@ -113,20 +113,62 @@ let check_hyps_inclusion env c sign =
 
 (* Make a type polymorphic if an arity *)
 
+let extract_level env p =
+  let _,c = dest_prod_assum env p in
+  match kind_of_term c with Sort (Type u) -> Some u | _ -> None
+
+let extract_context_levels env l =
+  let fold l (_, b, p) = match b with
+  | None -> extract_level env p :: l
+  | _ -> l
+  in
+  List.fold_left fold [] l
+
+let make_polymorphic_if_constant_for_ind env {uj_val = c; uj_type = t} =
+  let params, ccl = dest_prod_assum env t in
+  match kind_of_term ccl with
+  | Sort (Type u) when isInd (fst (decompose_app (whd_betadeltaiota env c))) ->
+      let param_ccls = extract_context_levels env params in
+      let s = { template_param_levels = param_ccls; template_level = u} in
+      TemplateArity (params,s)
+  | _ ->
+      RegularArity t
+
 (* Type of constants *)
 
-let type_of_constant env cst = constant_type env cst
-let type_of_constant_in env cst = constant_type_in env cst
-let type_of_constant_knowing_parameters env t _ = t
-let type_of_constant_type env cst = cst
+let type_of_constant_type_knowing_parameters env t paramtyps =
+  match t with
+  | RegularArity t -> t
+  | TemplateArity (sign,ar) ->
+      let ctx = List.rev sign in
+      let ctx,s = instantiate_universes env ctx ar paramtyps in
+      mkArity (List.rev ctx,s)
+
+let type_of_constant_knowing_parameters env cst paramtyps =
+  let ty, cu = constant_type env cst in
+    type_of_constant_type_knowing_parameters env ty paramtyps, cu
+
+let type_of_constant_type env t =
+  type_of_constant_type_knowing_parameters env t [||]
 
-let judge_of_constant env (kn,u as cst) =
+let type_of_constant env cst =
+  type_of_constant_knowing_parameters env cst [||]
+
+let type_of_constant_in env cst =
+  let ar = constant_type_in env cst in
+    type_of_constant_type_knowing_parameters env ar [||]
+
+let judge_of_constant_knowing_parameters env (kn,u as cst) jl =
   let c = mkConstU cst in
   let cb = lookup_constant kn env in
   let _ = check_hyps_inclusion env c cb.const_hyps in
-  let ty, cu = type_of_constant env cst in
+  let paramstyp = Array.map (fun j -> lazy j.uj_type) jl in
+  let ty, cu = type_of_constant_knowing_parameters env cst paramstyp in
   let _ = Environ.check_constraints cu env in
-    (make_judge c ty)
+    make_judge c ty
+
+let judge_of_constant env cst =
+  judge_of_constant_knowing_parameters env cst [||]
 
 let type_of_projection env (cst,u) =
   let cb = lookup_constant cst env in
diff --git a/kernel/typeops.mli b/kernel/typeops.mli
index 6bc1ce496..9747dbe83 100644
--- a/kernel/typeops.mli
+++ b/kernel/typeops.mli
@@ -51,10 +51,13 @@ val judge_of_relative : env -> int -> unsafe_judgment
 val judge_of_variable : env -> variable -> unsafe_judgment
 
 (** {6 type of a constant } *)
-val judge_of_constant : env -> constant puniverses -> unsafe_judgment
 
-(* val judge_of_constant_knowing_parameters : *)
-(*   env -> constant -> unsafe_judgment array -> unsafe_judgment *)
+val judge_of_constant : env -> pconstant -> unsafe_judgment
+
+val judge_of_constant_knowing_parameters :
+  env -> pconstant -> unsafe_judgment array -> unsafe_judgment
+
+(** {6 type of an applied projection } *)
 
 val judge_of_projection : env -> Names.projection -> unsafe_judgment -> unsafe_judgment
 
@@ -103,15 +106,20 @@ val judge_of_case : env -> case_info
 val type_fixpoint : env -> Name.t array -> types array
     -> unsafe_judgment array -> unit
 
-val type_of_constant : env -> constant puniverses -> types constrained
+val type_of_constant : env -> pconstant -> types constrained
 
 val type_of_constant_type : env -> constant_type -> types
 
 val type_of_projection : env -> Names.projection puniverses -> types
 
-val type_of_constant_in : env -> constant puniverses -> types
+val type_of_constant_in : env -> pconstant -> types
 
-val type_of_constant_knowing_parameters :
+val type_of_constant_type_knowing_parameters :
   env -> constant_type -> types Lazy.t array -> types
 
+val type_of_constant_knowing_parameters :
+  env -> pconstant -> types Lazy.t array -> types constrained
 
+(** Make a type polymorphic if an arity *)
+val make_polymorphic_if_constant_for_ind : env -> unsafe_judgment ->
+  constant_type
diff --git a/library/assumptions.ml b/library/assumptions.ml
index 9cfe531ce..37585caa4 100644
--- a/library/assumptions.ml
+++ b/library/assumptions.ml
@@ -222,7 +222,7 @@ let assumptions ?(add_opaque=false) ?(add_transparent=false) st (* t *) =
   and add_kn kn s acc =
     let cb = lookup_constant kn in
     let do_type cst =
-      let ctype = cb.Declarations.const_type in
+      let ctype = Global.type_of_global_unsafe (Globnames.ConstRef kn) in
 	(s,ContextObjectMap.add cst ctype acc)
     in
     let (s,acc) =
diff --git a/library/declare.ml b/library/declare.ml
index 820e72369..45015ac65 100644
--- a/library/declare.ml
+++ b/library/declare.ml
@@ -222,7 +222,8 @@ let declare_sideff se =
   in
   let ty_of cb =
     match cb.Declarations.const_type with
-    | (* Declarations.NonPolymorphicType  *)t -> Some t in
+    | Declarations.RegularArity t -> Some t 
+    | Declarations.TemplateArity _ -> None in
   let cst_of cb =
     let pt, opaque = pt_opaque_of cb in
     let ty = ty_of cb in
diff --git a/library/global.ml b/library/global.ml
index 16b07db25..fbba81b51 100644
--- a/library/global.ml
+++ b/library/global.ml
@@ -151,10 +151,16 @@ let type_of_global_unsafe r =
   match r with
   | VarRef id -> Environ.named_type id env
   | ConstRef c -> 
-     let cb = Environ.lookup_constant c env in cb.Declarations.const_type
+     let cb = Environ.lookup_constant c env in 
+       Typeops.type_of_constant_type env cb.Declarations.const_type
   | IndRef ind ->
-     let (mib, oib) = Inductive.lookup_mind_specif env ind in
-       oib.Declarations.mind_arity.Declarations.mind_user_arity
+     let (mib, oib as specif) = Inductive.lookup_mind_specif env ind in
+     let inst = 
+       if mib.Declarations.mind_polymorphic then 
+	 Univ.UContext.instance mib.Declarations.mind_universes
+       else Univ.Instance.empty
+     in
+       Inductive.type_of_inductive env (specif, inst)
   | ConstructRef cstr ->
      let (mib,oib as specif) = Inductive.lookup_mind_specif env (inductive_of_constructor cstr) in
      let inst = Univ.UContext.instance mib.Declarations.mind_universes in
@@ -169,13 +175,13 @@ let type_of_global_in_context env r =
      let univs = 
        if cb.const_polymorphic then Future.force cb.const_universes 
        else Univ.UContext.empty
-     in cb.Declarations.const_type, univs
+     in Typeops.type_of_constant_type env cb.Declarations.const_type, univs
   | IndRef ind ->
-     let (mib, oib) = Inductive.lookup_mind_specif env ind in
+     let (mib, oib as specif) = Inductive.lookup_mind_specif env ind in
      let univs = 
        if mib.mind_polymorphic then mib.mind_universes 
        else Univ.UContext.empty
-     in oib.Declarations.mind_arity.Declarations.mind_user_arity, univs
+     in Inductive.type_of_inductive env (specif, Univ.UContext.instance univs), univs
   | ConstructRef cstr ->
      let (mib,oib as specif) = Inductive.lookup_mind_specif env (inductive_of_constructor cstr) in
      let univs = 
diff --git a/library/impargs.ml b/library/impargs.ml
index 5a44b5bdb..f0292762f 100644
--- a/library/impargs.ml
+++ b/library/impargs.ml
@@ -403,7 +403,7 @@ let compute_semi_auto_implicits env f manual t =
 let compute_constant_implicits flags manual cst =
   let env = Global.env () in
   let cb = Environ.lookup_constant cst env in
-  let ty = cb.const_type in
+  let ty = Typeops.type_of_constant_type env cb.const_type in
   let impls = compute_semi_auto_implicits env flags manual ty in
     impls
     (* match cb.const_proj with *)
diff --git a/library/universes.ml b/library/universes.ml
index c7009b400..73a0b2b1c 100644
--- a/library/universes.ml
+++ b/library/universes.ml
@@ -144,17 +144,21 @@ let type_of_reference env r =
   | VarRef id -> Environ.named_type id env, ContextSet.empty
   | ConstRef c ->
      let cb = Environ.lookup_constant c env in
+     let ty = Typeops.type_of_constant_type env cb.const_type in
        if cb.const_polymorphic then
 	 let (inst, subst), ctx = fresh_instance_from (Future.force cb.const_universes) in
-	   Vars.subst_univs_constr subst cb.const_type, ctx
-       else cb.const_type, ContextSet.empty
+	   Vars.subst_univs_constr subst ty, ctx
+       else ty, ContextSet.empty
 
   | IndRef ind ->
-     let (mib, oib) = Inductive.lookup_mind_specif env ind in
+     let (mib, oib as specif) = Inductive.lookup_mind_specif env ind in
        if mib.mind_polymorphic then
 	 let (inst, subst), ctx = fresh_instance_from mib.mind_universes in
-	   Vars.subst_univs_constr subst oib.mind_arity.mind_user_arity, ctx
-       else oib.mind_arity.mind_user_arity, ContextSet.empty
+	 let ty = Inductive.type_of_inductive env (specif, inst) in
+	   ty, ctx
+       else
+	 let ty = Inductive.type_of_inductive env (specif, Univ.Instance.empty) in
+	   ty, ContextSet.empty
   | ConstructRef cstr ->
      let (mib,oib as specif) = Inductive.lookup_mind_specif env (inductive_of_constructor cstr) in
        if mib.mind_polymorphic then
diff --git a/plugins/Derive/g_derive.ml4 b/plugins/Derive/g_derive.ml4
index 9137c3d28..c7db26fb8 100644
--- a/plugins/Derive/g_derive.ml4
+++ b/plugins/Derive/g_derive.ml4
@@ -8,7 +8,7 @@
 
 (*i camlp4deps: "grammar/grammar.cma"  i*)
 
-let classify_derive_command _ = Vernacexpr.(VtStartProof ("Classic",Doesn'tGuaranteeOpacity,[]),VtLater)
+let classify_derive_command _ = Vernacexpr.(VtStartProof ("Classic",Doesn'tGuaranteeOpacity,[], false),VtLater)
 
 VERNAC COMMAND EXTEND Derive CLASSIFIED BY classify_derive_command
 | [ "Derive" ident(f) "From" constr(init) "Upto" constr(r) "As" ident(lemma) ] ->
diff --git a/plugins/extraction/extract_env.ml b/plugins/extraction/extract_env.ml
index 74de31368..791294902 100644
--- a/plugins/extraction/extract_env.ml
+++ b/plugins/extraction/extract_env.ml
@@ -131,7 +131,7 @@ end
 exception Impossible
 
 let check_arity env cb =
-  let t = cb.const_type in
+  let t = Typeops.type_of_constant_type env cb.const_type in
   if Reduction.is_arity env t then raise Impossible
 
 let check_fix env cb i =
diff --git a/plugins/extraction/extraction.ml b/plugins/extraction/extraction.ml
index 5b79f6d78..f7b677a1e 100644
--- a/plugins/extraction/extraction.ml
+++ b/plugins/extraction/extraction.ml
@@ -202,17 +202,15 @@ let parse_ind_args si args relmax =
 let oib_equal o1 o2 =
   Id.equal o1.mind_typename o2.mind_typename &&
   List.equal eq_rel_declaration o1.mind_arity_ctxt o2.mind_arity_ctxt &&
-  begin match o1.mind_arity, o2.mind_arity with
-  (* | Monomorphic {mind_user_arity=c1; mind_sort=s1}, *)
-  (*   Monomorphic {mind_user_arity=c2; mind_sort=s2} -> *)
-  (*     eq_constr c1 c2 && Sorts.equal s1 s2 *)
-  (* | ma1, ma2 -> Pervasives.(=) ma1 ma2 (\** FIXME: this one is surely wrong *\) end && *)
-  (*   Array.equal Id.equal o1.mind_consnames o2.mind_consnames *)
-  | {mind_user_arity=c1; mind_sort=s1},
-    {mind_user_arity=c2; mind_sort=s2} ->
-      eq_constr c1 c2 && Sorts.equal s1 s2
-  end &&
-  Array.equal Id.equal o1.mind_consnames o2.mind_consnames
+    begin
+      match o1.mind_arity, o2.mind_arity with
+      | RegularArity {mind_user_arity=c1; mind_sort=s1}, RegularArity {mind_user_arity=c2; mind_sort=s2} ->
+	eq_constr c1 c2 && Sorts.equal s1 s2
+      | {mind_user_arity=c1; mind_sort=s1},
+	{mind_user_arity=c2; mind_sort=s2} ->
+	eq_constr c1 c2 && Sorts.equal s1 s2
+    end &&
+    Array.equal Id.equal o1.mind_consnames o2.mind_consnames
 
 let mib_equal m1 m2 =
   Array.equal oib_equal m1.mind_packets m1.mind_packets &&
@@ -525,7 +523,8 @@ and mlt_env env r = match r with
 	   | _ -> None
        with Not_found ->
 	 let cb = Environ.lookup_constant kn env in
-	 let typ = cb.const_type (* FIXME not sure if we should instantiate univs here *) in
+	 let typ = Typeops.type_of_constant_type env cb.const_type
+ (* FIXME not sure if we should instantiate univs here *) in
 	 match cb.const_body with
 	   | Undef _ | OpaqueDef _ -> None
 	   | Def l_body ->
@@ -553,7 +552,7 @@ let record_constant_type env kn opt_typ =
     lookup_type kn
   with Not_found ->
     let typ = match opt_typ with
-      | None -> (lookup_constant kn env).const_type
+      | None -> Typeops.type_of_constant_type env (lookup_constant kn env).const_type
       | Some typ -> typ
     in let mlt = extract_type env [] 1 typ []
     in let schema = (type_maxvar mlt, mlt)
@@ -976,7 +975,7 @@ let extract_fixpoint env vkn (fi,ti,ci) =
 
 let extract_constant env kn cb =
   let r = ConstRef kn in
-  let typ = cb.const_type in
+  let typ = Global.type_of_global_unsafe r in
   let warn_info () = if not (is_custom r) then add_info_axiom r in
   let warn_log () = if not (constant_has_body cb) then add_log_axiom r
   in
@@ -1023,7 +1022,7 @@ let extract_constant env kn cb =
 
 let extract_constant_spec env kn cb =
   let r = ConstRef kn in
-  let typ = cb.const_type in
+  let typ = Global.type_of_global_unsafe r in
   match flag_of_type env typ with
     | (Logic, TypeScheme) -> Stype (r, [], Some (Tdummy Ktype))
     | (Logic, Default) -> Sval (r, Tdummy Kother)
diff --git a/plugins/extraction/table.ml b/plugins/extraction/table.ml
index 133f4ada9..c48873c80 100644
--- a/plugins/extraction/table.ml
+++ b/plugins/extraction/table.ml
@@ -816,7 +816,7 @@ let extract_constant_inline inline r ids s =
   match g with
     | ConstRef kn ->
 	let env = Global.env () in
-	let typ = (Environ.lookup_constant kn env).const_type in
+	let typ = Global.type_of_global_unsafe (ConstRef kn) in
 	let typ = Reduction.whd_betadeltaiota env typ in
 	if Reduction.is_arity env typ
 	  then begin
diff --git a/plugins/funind/functional_principles_proofs.ml b/plugins/funind/functional_principles_proofs.ml
index a3af23dcd..f0c7b5a7f 100644
--- a/plugins/funind/functional_principles_proofs.ml
+++ b/plugins/funind/functional_principles_proofs.ml
@@ -964,7 +964,7 @@ let generate_equation_lemma fnames f fun_num nb_params nb_args rec_args_num =
   let eq_rhs = nf_betaiotazeta (mkApp(compose_lam params f_body_with_params_and_other_fun,Array.init (nb_params + nb_args) (fun i -> mkRel(nb_params + nb_args - i)))) in
 (*   observe (str "eq_rhs " ++  pr_lconstr eq_rhs); *)
   let type_ctxt,type_of_f = decompose_prod_n_assum (nb_params + nb_args)
-    ((*FIXME*)f_def.const_type) in
+    (Typeops.type_of_constant_type (Global.env ()) (*FIXME*)f_def.const_type) in
   let eqn = mkApp(Lazy.force eq,[|type_of_f;eq_lhs;eq_rhs|]) in
   let lemma_type = it_mkProd_or_LetIn eqn type_ctxt in
   let f_id = Label.to_id (con_label (fst (destConst f))) in
diff --git a/plugins/funind/indfun.ml b/plugins/funind/indfun.ml
index d6ad5575b..49f833590 100644
--- a/plugins/funind/indfun.ml
+++ b/plugins/funind/indfun.ml
@@ -771,7 +771,7 @@ let make_graph (f_ref:global_reference) =
 	   with_full_print (fun () ->
 		(Constrextern.extern_constr false env body,
 		 Constrextern.extern_type false env
-                   ((*FIXNE*) c_body.const_type)
+                   ((*FIXNE*) Typeops.type_of_constant_type env c_body.const_type)
 		)
 	     )
 	     ()
diff --git a/plugins/funind/merge.ml b/plugins/funind/merge.ml
index d0497f6f6..2e524a35f 100644
--- a/plugins/funind/merge.ml
+++ b/plugins/funind/merge.ml
@@ -139,7 +139,7 @@ let showind (id:Id.t) =
     print_string (string_of_name nm^":");
     prconstr tp; print_string "\n")
     ib1.mind_arity_ctxt;
-    Printf.printf "arity :"; prconstr ib1.mind_arity.mind_user_arity;
+    Printf.printf "arity :"; prconstr (Inductiveops.type_of_inductive (Global.env ()) ind1);
   Array.iteri
     (fun i x -> Printf.printf"type constr %d :" i ; prconstr x)
     ib1.mind_user_lc
diff --git a/plugins/xml/xmlcommand.ml b/plugins/xml/xmlcommand.ml
index 3d655920b..8f47859da 100644
--- a/plugins/xml/xmlcommand.ml
+++ b/plugins/xml/xmlcommand.ml
@@ -380,7 +380,7 @@ let print internal glob_ref kind xml_library_root =
        let val0 = Declareops.body_of_constant cb in
        let typ = cb.Declarations.const_type in
        let hyps = cb.Declarations.const_hyps in
-       let typ = (* Typeops.type_of_constant_type (Global.env()) *) typ in
+       let typ = Typeops.type_of_constant_type (Global.env()) typ in
         Cic2acic.Constant kn,mk_constant_obj id val0 typ variables hyps
     | Globnames.IndRef (kn,_) ->
        let mib = Global.lookup_mind kn in
diff --git a/pretyping/inductiveops.ml b/pretyping/inductiveops.ml
index 7e4d37b2e..0b7cd89f2 100644
--- a/pretyping/inductiveops.ml
+++ b/pretyping/inductiveops.ml
@@ -451,9 +451,44 @@ let arity_of_case_predicate env (ind,params) dep k =
 (* Inferring the sort of parameters of a polymorphic inductive type
    knowing the sort of the conclusion *)
 
-let type_of_inductive_knowing_conclusion env ((mib,mip),u) conclty =
-  let subst = Inductive.make_inductive_subst mib u in
-    subst_univs_constr subst mip.mind_arity.mind_user_arity
+
+(* Compute the inductive argument types: replace the sorts
+   that appear in the type of the inductive by the sort of the
+   conclusion, and the other ones by fresh universes. *)
+let rec instantiate_universes env evdref scl is = function
+  | (_,Some _,_ as d)::sign, exp ->
+      d :: instantiate_universes env evdref scl is (sign, exp)
+  | d::sign, None::exp ->
+      d :: instantiate_universes env evdref scl is (sign, exp)
+  | (na,None,ty)::sign, Some u::exp ->
+      let ctx,_ = Reduction.dest_arity env ty in
+      let s =
+	(* Does the sort of parameter [u] appear in (or equal)
+           the sort of inductive [is] ? *)
+        if univ_depends u is then
+          scl (* constrained sort: replace by scl *)
+        else
+          (* unconstriained sort: replace by fresh universe *)
+          let evm, s = Evd.new_sort_variable Evd.univ_flexible !evdref in
+	    evdref := evm; s
+      in
+      (na,None,mkArity(ctx,s)):: instantiate_universes env evdref scl is (sign, exp)
+  | sign, [] -> sign (* Uniform parameters are exhausted *)
+  | [], _ -> assert false
+
+let type_of_inductive_knowing_conclusion env sigma ((mib,mip),u) conclty =
+  match mip.mind_arity with
+  | RegularArity s ->
+    let subst = Inductive.make_inductive_subst mib u in
+      sigma, subst_univs_constr subst s.mind_user_arity
+  | TemplateArity ar ->
+    let _,scl = Reduction.dest_arity env conclty in
+    let ctx = List.rev mip.mind_arity_ctxt in
+    let evdref = ref sigma in
+    let ctx =
+      instantiate_universes
+        env evdref scl ar.template_level (ctx,ar.template_param_levels) in
+      !evdref, mkArity (List.rev ctx,scl)
 
 (***********************************************)
 (* Guard condition *)
diff --git a/pretyping/inductiveops.mli b/pretyping/inductiveops.mli
index 39451ec05..34ddce72b 100644
--- a/pretyping/inductiveops.mli
+++ b/pretyping/inductiveops.mli
@@ -142,7 +142,7 @@ i*)
 (********************)
 
 val type_of_inductive_knowing_conclusion :
-  env -> Inductive.mind_specif puniverses -> types -> types
+  env -> evar_map -> Inductive.mind_specif puniverses -> types -> evar_map * types
 
 (********************)
 val control_only_guard : env -> types -> unit
diff --git a/pretyping/retyping.ml b/pretyping/retyping.ml
index 31487125a..ccb9420fd 100644
--- a/pretyping/retyping.ml
+++ b/pretyping/retyping.ml
@@ -190,7 +190,7 @@ let retype ?(polyprop=true) sigma =
 	 with Reduction.NotArity -> retype_error NotAnArity)
     | Const cst ->
       let t = constant_type_in env cst in
-	(try Typeops.type_of_constant_knowing_parameters env t argtyps
+	(try Typeops.type_of_constant_type_knowing_parameters env t argtyps
 	 with Reduction.NotArity -> retype_error NotAnArity)
     | Var id -> type_of_var env id
     | Construct cstr -> type_of_constructor env cstr
@@ -211,13 +211,13 @@ let type_of_global_reference_knowing_conclusion env sigma c conclty =
   match kind_of_term c with
     | Ind (ind,u) ->
         let spec = Inductive.lookup_mind_specif env ind in
-        type_of_inductive_knowing_conclusion env (spec,u) conclty
+          type_of_inductive_knowing_conclusion env sigma (spec,u) conclty
     | Const cst ->
         let t = constant_type_in env cst in
         (* TODO *)
-        Typeops.type_of_constant_knowing_parameters env t [||]
-    | Var id -> type_of_var env id
-    | Construct cstr -> type_of_constructor env cstr
+          sigma, Typeops.type_of_constant_type_knowing_parameters env t [||]
+    | Var id -> sigma, type_of_var env id
+    | Construct cstr -> sigma, type_of_constructor env cstr
     | _ -> assert false
 
 (* Profiling *)
diff --git a/pretyping/retyping.mli b/pretyping/retyping.mli
index fc1dd3564..b8b458555 100644
--- a/pretyping/retyping.mli
+++ b/pretyping/retyping.mli
@@ -41,4 +41,4 @@ val type_of_global_reference_knowing_parameters : env -> evar_map -> constr ->
   constr array -> types
 
 val type_of_global_reference_knowing_conclusion :
-  env -> evar_map -> constr -> types -> types
+  env -> evar_map -> constr -> types -> evar_map * types
diff --git a/pretyping/typing.ml b/pretyping/typing.ml
index bd559ddd5..cc473bd86 100644
--- a/pretyping/typing.ml
+++ b/pretyping/typing.ml
@@ -27,7 +27,7 @@ let meta_type evd mv =
 
 let constant_type_knowing_parameters env cst jl =
   let paramstyp = Array.map (fun j -> lazy j.uj_type) jl in
-  type_of_constant_knowing_parameters env (constant_type_in env cst) paramstyp
+  type_of_constant_type_knowing_parameters env (constant_type_in env cst) paramstyp
 
 let inductive_type_knowing_parameters env (ind,u) jl =
   let mspec = lookup_mind_specif env ind in
diff --git a/pretyping/vnorm.ml b/pretyping/vnorm.ml
index 16eeaa293..2cf730f11 100644
--- a/pretyping/vnorm.ml
+++ b/pretyping/vnorm.ml
@@ -98,7 +98,8 @@ let construct_of_constr_block = construct_of_constr false
 let constr_type_of_idkey env idkey =
   match idkey with
   | ConstKey cst ->
-      mkConst cst, (Environ.lookup_constant cst env).const_type
+    let const_type = (Environ.lookup_constant cst env).const_type in
+      mkConst cst, Typeops.type_of_constant_type env const_type
   | VarKey id ->
       let (_,_,ty) = lookup_named id env in
       mkVar id, ty
diff --git a/printing/prettyp.ml b/printing/prettyp.ml
index e885f5978..87d7e0980 100644
--- a/printing/prettyp.ml
+++ b/printing/prettyp.ml
@@ -371,7 +371,8 @@ let print_body = function
 let print_typed_body (val_0,typ) =
   (print_body val_0 ++ fnl () ++ str "     : " ++ pr_ltype typ)
 
-let ungeneralized_type_of_constant_type t = t
+let ungeneralized_type_of_constant_type t = 
+  Typeops.type_of_constant_type (Global.env ()) t
 
 let print_constant with_values sep sp =
   let cb = Global.lookup_constant sp in
diff --git a/printing/printer.ml b/printing/printer.ml
index 91156e21f..cb10f9661 100644
--- a/printing/printer.ml
+++ b/printing/printer.ml
@@ -755,16 +755,16 @@ let print_constructors envpar names types =
   hv 0 (str "  " ++ pc)
 
 let build_ind_type env mip =
-  mip.mind_arity.mind_user_arity
+  Inductive.type_of_inductive env mip
 
 let print_one_inductive env mib ((_,i) as ind) =
   let mip = mib.mind_packets.(i) in
   let params = mib.mind_params_ctxt in
   let args = extended_rel_list 0 params in
-  let arity = hnf_prod_applist env (build_ind_type env mip) args in
   let u = if mib.mind_polymorphic then 
       Univ.UContext.instance mib.mind_universes 
     else Univ.Instance.empty in
+  let arity = hnf_prod_applist env (build_ind_type env ((mib,mip),u)) args in
   let cstrtypes = Inductive.type_of_constructors (ind,u) (mib,mip) in
   let cstrtypes = Array.map (fun c -> hnf_prod_applist env c args) cstrtypes in
   let envpar = push_rel_context params env in
@@ -802,7 +802,7 @@ let print_record env mind mib =
   let u = if mib.mind_polymorphic then 
       Univ.UContext.instance mib.mind_universes 
     else Univ.Instance.empty in
-  let arity = hnf_prod_applist env (build_ind_type env mip) args in
+  let arity = hnf_prod_applist env (build_ind_type env ((mib,mip),u)) args in
   let cstrtypes = Inductive.type_of_constructors ((mind,0),u) (mib,mip) in
   let cstrtype = hnf_prod_applist env cstrtypes.(0) args in
   let fields = get_fields cstrtype in
diff --git a/printing/printmod.ml b/printing/printmod.ml
index da5546bac..8d05e2d6d 100644
--- a/printing/printmod.ml
+++ b/printing/printmod.ml
@@ -146,7 +146,8 @@ let print_body is_impl env mp (l,body) =
 	  | None -> mt ()
 	  | Some env ->
 	    str " :" ++ spc () ++
-	    hov 0 (Printer.pr_ltype_env env cb.const_type) ++
+	    hov 0 (Printer.pr_ltype_env env 
+		     (Typeops.type_of_constant_type env cb.const_type)) ++
 	    (match cb.const_body with
 	      | Def l when is_impl ->
 		spc () ++
diff --git a/proofs/logic.ml b/proofs/logic.ml
index 02f3a16d8..fb88b8754 100644
--- a/proofs/logic.ml
+++ b/proofs/logic.ml
@@ -382,10 +382,11 @@ let rec mk_refgoals sigma goal goalacc conclty trm =
 	  match kind_of_term f with
 	    | Ind _ | Const _
 		when (isInd f || has_polymorphic_type (fst (destConst f))) ->
+	      let sigma, ty = 
 		(* Sort-polymorphism of definition and inductive types *)
-		goalacc,
-                type_of_global_reference_knowing_conclusion env sigma f conclty,
-		sigma, f
+		type_of_global_reference_knowing_conclusion env sigma f conclty
+	      in
+		goalacc, ty, sigma, f
 	    | _ ->
 		mk_hdgoals sigma goal goalacc f
 	in
diff --git a/toplevel/discharge.ml b/toplevel/discharge.ml
index 55475a378..e17038a4f 100644
--- a/toplevel/discharge.ml
+++ b/toplevel/discharge.ml
@@ -69,23 +69,34 @@ let abstract_inductive hyps nparams inds =
   in (params',ind'')
 
 let refresh_polymorphic_type_of_inductive (_,mip) =
-  mip.mind_arity.mind_user_arity
+  match mip.mind_arity with
+  | RegularArity s -> s.mind_user_arity, Univ.ContextSet.empty
+  | TemplateArity ar ->
+    let ctx = List.rev mip.mind_arity_ctxt in
+    let univ, uctx = Universes.new_global_univ () in
+      mkArity (List.rev ctx, Type univ), uctx
 
 let process_inductive (sechyps,abs_ctx) modlist mib =
   let nparams = mib.mind_nparams in
+  let univctx = ref Univ.ContextSet.empty in
   let inds =
     Array.map_to_list
       (fun mip ->
-	 let arity = expmod_constr modlist (refresh_polymorphic_type_of_inductive (mib,mip)) in
-	 let lc = Array.map (expmod_constr modlist) mip.mind_user_lc in
-	 (mip.mind_typename,
-	  arity,
-	  Array.to_list mip.mind_consnames,
-	  Array.to_list lc))
+	let ty, uctx = refresh_polymorphic_type_of_inductive (mib,mip) in
+	let () = univctx := Univ.ContextSet.union uctx !univctx in
+	let arity = expmod_constr modlist ty in
+	let lc = Array.map (expmod_constr modlist) mip.mind_user_lc in
+	  (mip.mind_typename,
+	   arity,
+	   Array.to_list mip.mind_consnames,
+	   Array.to_list lc))
       mib.mind_packets in
   let sechyps' = map_named_context (expmod_constr modlist) sechyps in
   let (params',inds') = abstract_inductive sechyps' nparams inds in
-  let univs = Univ.UContext.union abs_ctx mib.mind_universes in
+  let univs = Univ.UContext.union abs_ctx 
+    (Univ.UContext.union (Univ.ContextSet.to_context !univctx) 
+       mib.mind_universes)
+  in
   { mind_entry_record = mib.mind_record <> None;
     mind_entry_finite = mib.mind_finite;
     mind_entry_params = params';
diff --git a/toplevel/search.ml b/toplevel/search.ml
index 1535ae617..37403504d 100644
--- a/toplevel/search.ml
+++ b/toplevel/search.ml
@@ -60,14 +60,15 @@ let iter_declarations (fn : global_reference -> env -> constr -> unit) =
     with Not_found -> (* we are in a section *) () end
   | "CONSTANT" ->
     let cst = Global.constant_of_delta_kn kn in
-    let typ, _ = Environ.constant_type_in_ctx env cst in
-    fn (ConstRef cst) env typ
+    let gr = ConstRef cst in
+    let typ = Global.type_of_global_unsafe gr in
+      fn gr env typ
   | "INDUCTIVE" ->
     let mind = Global.mind_of_delta_kn kn in
     let mib = Global.lookup_mind mind in
     let iter_packet i mip =
       let ind = (mind, i) in
-      let i = (ind, Univ.UContext.instance mib.mind_universes) in
+      let i = (ind, Inductive.inductive_instance mib) in
       let typ = Inductiveops.type_of_inductive env i in
       let () = fn (IndRef ind) env typ in
       let len = Array.length mip.mind_user_lc in
diff --git a/toplevel/vernacentries.ml b/toplevel/vernacentries.ml
index 02e59a227..67ca1b666 100644
--- a/toplevel/vernacentries.ml
+++ b/toplevel/vernacentries.ml
@@ -252,7 +252,8 @@ let print_namespace ns =
     print_list pr_id qn
   in
   let print_constant k body =
-    let t = body.Declarations.const_type in
+    (* FIXME: universes *)
+    let t = Typeops.type_of_constant_type (Global.env ()) body.Declarations.const_type in
     print_kn k ++ str":" ++ spc() ++ Printer.pr_type t
   in
   let matches mp = match match_modulepath ns mp with
-- 
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