Merge PR #79: Let the kernel assume that a (co-)inductive type is positive.

38 files changed, 387 insertions, 226 deletions

diff --git a/checker/cic.mli b/checker/cic.mli
index 00ac2f56c..469cf8d4c 100644
--- a/checker/cic.mli
+++ b/checker/cic.mli
@@ -212,6 +212,15 @@ type constant_def =
 
 type constant_universes = Univ.universe_context
 
+(** The [typing_flags] are instructions to the type-checker which
+    modify its behaviour. The typing flags used in the type-checking
+    of a constant are tracked in their {!constant_body} so that they
+    can be displayed to the user. *)
+type typing_flags = {
+  check_guarded : bool; (** If [false] then fixed points and co-fixed
+                            points are assumed to be total. *)
+}
+
 type constant_body = {
     const_hyps : section_context; (** New: younger hyp at top *)
     const_body : constant_def;
@@ -220,7 +229,9 @@ type constant_body = {
     const_polymorphic : bool; (** Is it polymorphic or not *)
     const_universes : constant_universes;
     const_proj : projection_body option;
-    const_inline_code : bool }
+    const_inline_code : bool;
+    const_typing_flags : typing_flags;
+}
 
 (** {6 Representation of mutual inductive types } *)
 
@@ -316,6 +327,8 @@ type mutual_inductive_body = {
 
     mind_private : bool option; (** allow pattern-matching: Some true ok, Some false blocked *)
 
+    mind_checked_positive : bool; (** [false] when the mutual-inductive was assumed to be well-founded, bypassing the positivity checker.  *)
+
 (** {8 Data for native compilation } *)
 
     mind_native_name : native_name ref; (** status of the code (linked or not, and where) *)
diff --git a/checker/values.ml b/checker/values.ml
index 19cbb5060..dd29f6fbe 100644
--- a/checker/values.ml
+++ b/checker/values.ml
@@ -13,7 +13,7 @@
 To ensure this file is up-to-date, 'make' now compares the md5 of cic.mli
 with a copy we maintain here:
 
-MD5 9f7fd499f812b6548a55f7067e6a9d06 checker/cic.mli
+MD5 7d7963269852d32324e10aa77beb938d  checker/cic.mli
 
 *)
 
@@ -213,6 +213,9 @@ let v_projbody =
 	    v_tuple "proj_eta" [|v_constr;v_constr|];
 	    v_constr|]
 
+let v_typing_flags =
+  v_tuple "typing_flags" [|v_bool|]
+
 let v_cb = v_tuple "constant_body"
   [|v_section_ctxt;
     v_cst_def;
@@ -221,7 +224,8 @@ let v_cb = v_tuple "constant_body"
     v_bool;
     v_context;
     Opt v_projbody;
-    v_bool|]
+    v_bool;
+    v_typing_flags|]
 
 let v_recarg = v_sum "recarg" 1 (* Norec *)
   [|[|v_ind|] (* Mrec *);[|v_ind|] (* Imbr *)|]
@@ -270,7 +274,8 @@ let v_ind_pack = v_tuple "mutual_inductive_body"
     v_rctxt;
     v_bool;
     v_context;
-    Opt v_bool|]
+    Opt v_bool;
+    v_bool|]
 
 let v_with =
   Sum ("with_declaration_body",0,
diff --git a/kernel/declarations.mli b/kernel/declarations.mli
index 1b77d5b7c..8b42a90e4 100644
--- a/kernel/declarations.mli
+++ b/kernel/declarations.mli
@@ -66,6 +66,15 @@ type constant_def =
 
 type constant_universes = Univ.universe_context
 
+(** The [typing_flags] are instructions to the type-checker which
+    modify its behaviour. The typing flags used in the type-checking
+    of a constant are tracked in their {!constant_body} so that they
+    can be displayed to the user. *)
+type typing_flags = {
+  check_guarded : bool; (** If [false] then fixed points and co-fixed
+                            points are assumed to be total. *)
+}
+
 (* some contraints are in constant_constraints, some other may be in
  * the OpaueDef *)
 type constant_body = {
@@ -76,7 +85,11 @@ type constant_body = {
     const_polymorphic : bool; (** Is it polymorphic or not *)
     const_universes : constant_universes;
     const_proj : projection_body option;
-    const_inline_code : bool }
+    const_inline_code : bool;
+    const_typing_flags : typing_flags; (** The typing options which
+                                           were used for
+                                           type-checking. *)
+}
 
 (** {6 Representation of mutual inductive types in the kernel } *)
 
@@ -178,6 +191,7 @@ type mutual_inductive_body = {
 
     mind_private : bool option; (** allow pattern-matching: Some true ok, Some false blocked *)
   
+    mind_checked_positive : bool; (** [false] when the mutual-inductive was assumed to be well-founded, bypassing the positivity checker.  *)
 }
 
 (** {6 Module declarations } *)
diff --git a/kernel/declareops.ml b/kernel/declareops.ml
index a09a8b786..78e2f386e 100644
--- a/kernel/declareops.ml
+++ b/kernel/declareops.ml
@@ -131,7 +131,8 @@ let subst_const_body sub cb =
           Option.map (Cemitcodes.subst_to_patch_subst sub) cb.const_body_code;
         const_polymorphic = cb.const_polymorphic;
         const_universes = cb.const_universes;
-        const_inline_code = cb.const_inline_code }
+        const_inline_code = cb.const_inline_code;
+        const_typing_flags = cb.const_typing_flags }
 
 (** {7 Hash-consing of constants } *)
 
@@ -254,7 +255,9 @@ let subst_mind_body sub mib =
     mind_packets = Array.smartmap (subst_mind_packet sub) mib.mind_packets ;
     mind_polymorphic = mib.mind_polymorphic;
     mind_universes = mib.mind_universes;
-    mind_private = mib.mind_private }
+    mind_private = mib.mind_private;
+    mind_checked_positive = mib.mind_checked_positive;
+  }
 
 let inductive_instance mib =
   if mib.mind_polymorphic then
diff --git a/kernel/entries.mli b/kernel/entries.mli
index d07ca2103..8b29e3abd 100644
--- a/kernel/entries.mli
+++ b/kernel/entries.mli
@@ -51,7 +51,10 @@ type mutual_inductive_entry = {
   mind_entry_inds : one_inductive_entry list;
   mind_entry_polymorphic : bool; 
   mind_entry_universes : Univ.universe_context;
-  mind_entry_private : bool option }
+  mind_entry_private : bool option;
+  mind_entry_check_positivity : bool;
+  (** [false] if positivity is to be assumed. *)
+}
 
 (** {6 Constants (Definition/Axiom) } *)
 type 'a proof_output = constr Univ.in_universe_context_set * 'a
diff --git a/kernel/fast_typeops.ml b/kernel/fast_typeops.ml
index 7f4ba8ecb..35c162cf3 100644
--- a/kernel/fast_typeops.ml
+++ b/kernel/fast_typeops.ml
@@ -327,7 +327,7 @@ let type_fixpoint env lna lar vdef vdeft =
     (* ATTENTION : faudra faire le typage du contexte des Const,
     Ind et Constructsi un jour cela devient des constructions
     arbitraires et non plus des variables *)
-let rec execute env cstr =
+let rec execute ~flags env cstr =
   let open Context.Rel.Declaration in
   match kind_of_term cstr with
     (* Atomic terms *)
@@ -347,12 +347,12 @@ let rec execute env cstr =
       judge_of_constant env c
 	
     | Proj (p, c) ->
-        let ct = execute env c in
+        let ct = execute ~flags env c in
           judge_of_projection env p c ct
 
     (* Lambda calculus operators *)
     | App (f,args) ->
-        let argst = execute_array env args in
+        let argst = execute_array ~flags env args in
 	let ft =
 	  match kind_of_term f with
 	  | Ind ind when Environ.template_polymorphic_pind ind env ->
@@ -365,7 +365,7 @@ let rec execute env cstr =
 	      judge_of_constant_knowing_parameters env cst args
 	  | _ ->
 	    (* Full or no sort-polymorphism *)
-	    execute env f
+            execute ~flags env f
 	in
 
 	  judge_of_apply env f ft args argst
@@ -373,25 +373,25 @@ let rec execute env cstr =
     | Lambda (name,c1,c2) ->
       let _ = execute_is_type env c1 in
       let env1 = push_rel (LocalAssum (name,c1)) env in
-      let c2t = execute env1 c2 in
+      let c2t = execute ~flags env1 c2 in
         judge_of_abstraction env name c1 c2t
 
     | Prod (name,c1,c2) ->
-      let vars = execute_is_type env c1 in
+      let vars = execute_is_type ~flags env c1 in
       let env1 = push_rel (LocalAssum (name,c1)) env in
-      let vars' = execute_is_type env1 c2 in
+      let vars' = execute_is_type ~flags env1 c2 in
 	judge_of_product env name vars vars'
 
     | LetIn (name,c1,c2,c3) ->
-      let c1t = execute env c1 in
+      let c1t = execute ~flags env c1 in
       let _c2s = execute_is_type env c2 in
       let _ = judge_of_cast env c1 c1t DEFAULTcast c2 in
       let env1 = push_rel (LocalDef (name,c1,c2)) env in
-      let c3t = execute env1 c3 in
+      let c3t = execute ~flags env1 c3 in
 	subst1 c1 c3t
 
     | Cast (c,k,t) ->
-      let ct = execute env c in
+      let ct = execute ~flags env c in
       let _ts = execute_type env t in
       let _ = judge_of_cast env c ct k t in
 	t
@@ -404,20 +404,20 @@ let rec execute env cstr =
       judge_of_constructor env c
 
     | Case (ci,p,c,lf) ->
-        let ct = execute env c in
-        let pt = execute env p in
-        let lft = execute_array env lf in
+        let ct = execute ~flags env c in
+        let pt = execute ~flags env p in
+        let lft = execute_array ~flags env lf in
           judge_of_case env ci p pt c ct lf lft
 
     | Fix ((vn,i as vni),recdef) ->
-      let (fix_ty,recdef') = execute_recdef env recdef i in
+      let (fix_ty,recdef') = execute_recdef ~flags env recdef i in
       let fix = (vni,recdef') in
-        check_fix env fix; fix_ty
+        check_fix env ~flags fix; fix_ty
 	  
     | CoFix (i,recdef) ->
-      let (fix_ty,recdef') = execute_recdef env recdef i in
+      let (fix_ty,recdef') = execute_recdef ~flags env recdef i in
       let cofix = (i,recdef') in
-        check_cofix env cofix; fix_ty
+        check_cofix env ~flags cofix; fix_ty
 	  
     (* Partial proofs: unsupported by the kernel *)
     | Meta _ ->
@@ -426,38 +426,41 @@ let rec execute env cstr =
     | Evar _ ->
 	anomaly (Pp.str "the kernel does not support existential variables")
 
-and execute_is_type env constr =
-  let t = execute env constr in
+and execute_is_type ~flags env constr =
+  let t = execute ~flags env constr in
     check_type env constr t
 
-and execute_type env constr =
-  let t = execute env constr in
+and execute_type ~flags env constr =
+  let t = execute ~flags env constr in
     type_judgment env constr t
 
-and execute_recdef env (names,lar,vdef) i =
-  let lart = execute_array env lar in
+and execute_recdef ~flags env (names,lar,vdef) i =
+  let lart = execute_array ~flags env lar in
   let lara = Array.map2 (assumption_of_judgment env) lar lart in
   let env1 = push_rec_types (names,lara,vdef) env in
-  let vdeft = execute_array env1 vdef in
+  let vdeft = execute_array ~flags env1 vdef in
   let () = type_fixpoint env1 names lara vdef vdeft in
     (lara.(i),(names,lara,vdef))
 
-and execute_array env = Array.map (execute env)
+and execute_array ~flags env = Array.map (execute ~flags env)
 
 (* Derived functions *)
-let infer env constr =
-  let t = execute env constr in
+let infer ~flags env constr =
+  let t = execute ~flags env constr in
     make_judge constr t
 
 let infer = 
   if Flags.profile then
     let infer_key = Profile.declare_profile "Fast_infer" in
-      Profile.profile2 infer_key infer
-  else infer
+      Profile.profile3 infer_key (fun a b c -> infer ~flags:a b c)
+  else (fun a b c -> infer ~flags:a b c)
 
-let infer_type env constr =
-  execute_type env constr
+(* Restores the labels of [infer] lost to profiling. *)
+let infer ~flags env t = infer flags env t
 
-let infer_v env cv =
-  let jv = execute_array env cv in
+let infer_type ~flags env constr =
+  execute_type ~flags env constr
+
+let infer_v ~flags env cv =
+  let jv = execute_array ~flags env cv in
     make_judgev cv jv
diff --git a/kernel/fast_typeops.mli b/kernel/fast_typeops.mli
index 05d52b2d3..45a603038 100644
--- a/kernel/fast_typeops.mli
+++ b/kernel/fast_typeops.mli
@@ -8,6 +8,7 @@
 
 open Term
 open Environ
+open Declarations
 
 (** {6 Typing functions (not yet tagged as safe) }
     
@@ -18,6 +19,6 @@ open Environ
  *)
 
 
-val infer      : env -> constr       -> unsafe_judgment
-val infer_v    : env -> constr array -> unsafe_judgment array
-val infer_type : env -> types        -> unsafe_type_judgment
+val infer      : flags:typing_flags -> env -> constr       -> unsafe_judgment
+val infer_v    : flags:typing_flags -> env -> constr array -> unsafe_judgment array
+val infer_type : flags:typing_flags -> env -> types        -> unsafe_type_judgment
diff --git a/kernel/indtypes.ml b/kernel/indtypes.ml
index edb758f07..b74788d21 100644
--- a/kernel/indtypes.ml
+++ b/kernel/indtypes.ml
@@ -483,8 +483,12 @@ let array_min nmr a = if Int.equal nmr 0 then 0 else
     for use by the guard condition (terms at these positions are
     considered sub-terms) as well as the number of of non-uniform
     arguments (used to generate induction schemes, so a priori less
-    relevant to the kernel). *)
-let check_positivity_one recursive (env,_,ntypes,_ as ienv) paramsctxt (_,i as ind) nnonrecargs lcnames indlc =
+    relevant to the kernel).
+
+    If [chkpos] is [false] then positivity is assumed, and
+    [check_positivity_one] computes the subterms occurrences in a
+    best-effort fashion. *)
+let check_positivity_one ~chkpos recursive (env,_,ntypes,_ as ienv) paramsctxt (_,i as ind) nnonrecargs lcnames indlc =
   let nparamsctxt = Context.Rel.length paramsctxt in
   let nmr = Context.Rel.nhyps paramsctxt in
   (** Positivity of one argument [c] of a constructor (i.e. the
@@ -501,9 +505,12 @@ let check_positivity_one recursive (env,_,ntypes,_ as ienv) paramsctxt (_,i as i
                 recursive call. Occurrences in the right-hand side of
                 the product must be strictly positive.*)
             (match weaker_noccur_between env n ntypes b with
-		None -> failwith_non_pos_list n ntypes [b]
+	      | None when chkpos ->
+                  failwith_non_pos_list n ntypes [b]
+              | None ->
+                  check_pos (ienv_push_var ienv (na, b, mk_norec)) nmr d
               | Some b ->
-	          check_pos (ienv_push_var ienv (na, b, mk_norec)) nmr d)
+                  check_pos (ienv_push_var ienv (na, b, mk_norec)) nmr d)
 	| Rel k ->
             (try let (ra,rarg) = List.nth ra_env (k-1) in
             let largs = List.map (whd_betadeltaiota env) largs in
@@ -515,7 +522,8 @@ let check_positivity_one recursive (env,_,ntypes,_ as ienv) paramsctxt (_,i as i
               (** The case where one of the inductives of the mutually
                   inductive block occurs as an argument of another is not
                   known to be safe. So Coq rejects it. *)
-	      if not (List.for_all (noccur_between n ntypes) largs)
+	      if chkpos &&
+                 not (List.for_all (noccur_between n ntypes) largs)
 	      then failwith_non_pos_list n ntypes largs
 	      else (nmr1,rarg)
               with Failure _ | Invalid_argument _ -> (nmr,mk_norec))
@@ -530,8 +538,9 @@ let check_positivity_one recursive (env,_,ntypes,_ as ienv) paramsctxt (_,i as i
             (** If an inductive of the mutually inductive block
                 appears in any other way, then the positivy check gives
                 up. *)
-	    if noccur_between n ntypes x &&
-              List.for_all (noccur_between n ntypes) largs
+	    if not chkpos ||
+              (noccur_between n ntypes x &&
+               List.for_all (noccur_between n ntypes) largs)
 	    then (nmr,mk_norec)
 	    else failwith_non_pos_list n ntypes (x::largs)
 
@@ -553,7 +562,7 @@ let check_positivity_one recursive (env,_,ntypes,_ as ienv) paramsctxt (_,i as i
       (** Inductives of the inductive block being defined are only
           allowed to appear nested in the parameters of another inductive
           type. Not in the proper indices. *)
-      if not (List.for_all (noccur_between n ntypes) auxnonrecargs) then
+      if chkpos && not (List.for_all (noccur_between n ntypes) auxnonrecargs) then
 	failwith_non_pos_list n ntypes auxnonrecargs;
       (* Nested mutual inductive types are not supported *)
       let auxntyp = mib.mind_ntypes in
@@ -613,7 +622,8 @@ let check_positivity_one recursive (env,_,ntypes,_ as ienv) paramsctxt (_,i as i
                 | _ -> raise (IllFormedInd (LocalNotConstructor(paramsctxt,nnonrecargs)))
                 end
               else
-                if not (List.for_all (noccur_between n ntypes) largs)
+                if chkpos &&
+                   not (List.for_all (noccur_between n ntypes) largs)
                 then failwith_non_pos_list n ntypes largs
             in
             (nmr, List.rev lrec)
@@ -633,9 +643,13 @@ let check_positivity_one recursive (env,_,ntypes,_ as ienv) paramsctxt (_,i as i
   and nmr' = array_min nmr irecargs_nmr
   in (nmr', mk_paths (Mrec ind) irecargs)
 
-(** [check_positivity kn env_ar paramsctxt inds] checks that the mutually
-    inductive block [inds] is strictly positive. *)
-let check_positivity kn env_ar_par paramsctxt finite inds =
+(** [check_positivity ~chkpos kn env_ar paramsctxt inds] checks that the mutually
+    inductive block [inds] is strictly positive.
+
+    If [chkpos] is [false] then positivity is assumed, and
+    [check_positivity_one] computes the subterms occurrences in a
+    best-effort fashion. *)
+let check_positivity ~chkpos kn env_ar_par paramsctxt finite inds =
   let ntypes = Array.length inds in
   let recursive = finite != Decl_kinds.BiFinite in
   let rc = Array.mapi (fun j t -> (Mrec (kn,j),t)) (Rtree.mk_rec_calls ntypes) in
@@ -647,7 +661,7 @@ let check_positivity kn env_ar_par paramsctxt finite inds =
       List.init nparamsctxt (fun _ -> (Norec,mk_norec)) @ ra_env_ar in
     let ienv = (env_ar_par, 1+nparamsctxt, ntypes, ra_env_ar_par) in
     let nnonrecargs = Context.Rel.nhyps sign - nmr in
-    check_positivity_one recursive ienv paramsctxt (kn,i) nnonrecargs lcnames lc
+    check_positivity_one ~chkpos recursive ienv paramsctxt (kn,i) nnonrecargs lcnames lc
   in
   let irecargs_nmr = Array.mapi check_one inds in
   let irecargs = Array.map snd irecargs_nmr
@@ -802,7 +816,7 @@ let compute_projections ((kn, _ as ind), u as indu) n x nparamargs params
     Array.of_list (List.rev kns),
     Array.of_list (List.rev pbs)
 
-let build_inductive env p prv ctx env_ar paramsctxt kn isrecord isfinite inds nmr recargs =
+let build_inductive env p prv ctx env_ar paramsctxt kn isrecord isfinite inds nmr recargs is_checked =
   let ntypes = Array.length inds in
   (* Compute the set of used section variables *)
   let hyps = used_section_variables env inds in
@@ -908,6 +922,7 @@ let build_inductive env p prv ctx env_ar paramsctxt kn isrecord isfinite inds nm
       mind_polymorphic = p;
       mind_universes = ctx;
       mind_private = prv;
+      mind_checked_positive = is_checked;
     }
 
 (************************************************************************)
@@ -916,10 +931,11 @@ let build_inductive env p prv ctx env_ar paramsctxt kn isrecord isfinite inds nm
 let check_inductive env kn mie =
   (* First type-check the inductive definition *)
   let (env_ar, env_ar_par, paramsctxt, inds) = typecheck_inductive env mie in
+  let chkpos = mie.mind_entry_check_positivity in
   (* Then check positivity conditions *)
-  let (nmr,recargs) = check_positivity kn env_ar_par paramsctxt mie.mind_entry_finite inds in
+  let (nmr,recargs) = check_positivity ~chkpos kn env_ar_par paramsctxt mie.mind_entry_finite inds in
   (* Build the inductive packets *)
     build_inductive env mie.mind_entry_polymorphic mie.mind_entry_private
       mie.mind_entry_universes
       env_ar paramsctxt kn mie.mind_entry_record mie.mind_entry_finite
-      inds nmr recargs
+      inds nmr recargs chkpos
diff --git a/kernel/inductive.ml b/kernel/inductive.ml
index 499cbf0df..24bdaa5c4 100644
--- a/kernel/inductive.ml
+++ b/kernel/inductive.ml
@@ -1067,21 +1067,24 @@ let inductive_of_mutfix env ((nvect,bodynum),(names,types,bodies as recdef)) =
   (Array.map fst rv, Array.map snd rv)
 
 
-let check_fix env ((nvect,_),(names,_,bodies as recdef) as fix) =
-  let (minds, rdef) = inductive_of_mutfix env fix in
-  let get_tree (kn,i) =
-    let mib = Environ.lookup_mind kn env in
-    mib.mind_packets.(i).mind_recargs
-  in
-  let trees = Array.map (fun (mind,_) -> get_tree mind) minds in
-  for i = 0 to Array.length bodies - 1 do
-    let (fenv,body) = rdef.(i) in
-    let renv = make_renv fenv nvect.(i) trees.(i) in
-    try check_one_fix renv nvect trees body
-    with FixGuardError (fixenv,err) ->
-      error_ill_formed_rec_body fixenv err names i
-	(push_rec_types recdef env) (judgment_of_fixpoint recdef)
-  done
+let check_fix env ~flags ((nvect,_),(names,_,bodies as recdef) as fix) =
+  if flags.check_guarded then
+    let (minds, rdef) = inductive_of_mutfix env fix in
+    let get_tree (kn,i) =
+      let mib = Environ.lookup_mind kn env in
+      mib.mind_packets.(i).mind_recargs
+    in
+    let trees = Array.map (fun (mind,_) -> get_tree mind) minds in
+    for i = 0 to Array.length bodies - 1 do
+      let (fenv,body) = rdef.(i) in
+      let renv = make_renv fenv nvect.(i) trees.(i) in
+      try check_one_fix renv nvect trees body
+      with FixGuardError (fixenv,err) ->
+        error_ill_formed_rec_body fixenv err names i
+	  (push_rec_types recdef env) (judgment_of_fixpoint recdef)
+    done
+  else
+    ()
 
 (*
 let cfkey = Profile.declare_profile "check_fix";;
@@ -1192,12 +1195,15 @@ let check_one_cofix env nbfix def deftype =
 (* The  function which checks that the whole block of definitions
    satisfies the guarded condition *)
 
-let check_cofix env (bodynum,(names,types,bodies as recdef)) =
-  let nbfix = Array.length bodies in
-  for i = 0 to nbfix-1 do
-    let fixenv = push_rec_types recdef env in
-    try check_one_cofix fixenv nbfix bodies.(i) types.(i)
-    with CoFixGuardError (errenv,err) ->
-      error_ill_formed_rec_body errenv err names i
-	fixenv (judgment_of_fixpoint recdef)
-  done
+let check_cofix env ~flags (bodynum,(names,types,bodies as recdef)) =
+  if flags.check_guarded then
+    let nbfix = Array.length bodies in
+    for i = 0 to nbfix-1 do
+      let fixenv = push_rec_types recdef env in
+      try check_one_cofix fixenv nbfix bodies.(i) types.(i)
+      with CoFixGuardError (errenv,err) ->
+        error_ill_formed_rec_body errenv err names i
+	  fixenv (judgment_of_fixpoint recdef)
+    done
+  else
+    ()
diff --git a/kernel/inductive.mli b/kernel/inductive.mli
index c0d18bc6e..25a557472 100644
--- a/kernel/inductive.mli
+++ b/kernel/inductive.mli
@@ -94,8 +94,11 @@ val inductive_sort_family : one_inductive_body -> sorts_family
 val check_case_info : env -> pinductive -> case_info -> unit
 
 (** {6 Guard conditions for fix and cofix-points. } *)
-val check_fix : env -> fixpoint -> unit
-val check_cofix : env -> cofixpoint -> unit
+
+(** When [chk] is false, the guard condition is not actually
+    checked. *)
+val check_fix : env -> flags:typing_flags -> fixpoint -> unit
+val check_cofix : env -> flags:typing_flags -> cofixpoint -> unit
 
 (** {6 Support for sort-polymorphic inductive types } *)
 
diff --git a/kernel/safe_typing.ml b/kernel/safe_typing.ml
index ce05190b6..72d6ee518 100644
--- a/kernel/safe_typing.ml
+++ b/kernel/safe_typing.ml
@@ -373,8 +373,8 @@ let safe_push_named d env =
   Environ.push_named d env
 
 
-let push_named_def (id,de) senv =
-  let c,typ,univs = Term_typing.translate_local_def senv.revstruct senv.env id de in
+let push_named_def ~flags (id,de) senv =
+  let c,typ,univs = Term_typing.translate_local_def ~flags senv.revstruct senv.env id de in
   let poly = de.Entries.const_entry_polymorphic in
   let univs = Univ.ContextSet.of_context univs in
   let c, univs = match c with
@@ -388,9 +388,9 @@ let push_named_def (id,de) senv =
   let env'' = safe_push_named (LocalDef (id,c,typ)) senv'.env in
     univs, {senv' with env=env''}
 
-let push_named_assum ((id,t,poly),ctx) senv =
+let push_named_assum ~flags ((id,t,poly),ctx) senv =
   let senv' = push_context_set poly ctx senv in
-  let t = Term_typing.translate_local_assum senv'.env t in
+  let t = Term_typing.translate_local_assum ~flags senv'.env t in
   let env'' = safe_push_named (LocalAssum (id,t)) senv'.env in
     {senv' with env=env''}
 
@@ -479,7 +479,7 @@ let update_resolver f senv = { senv with modresolver = f senv.modresolver }
 
 (** Insertion of constants and parameters in environment *)
 type global_declaration =
-  | ConstantEntry of bool * private_constants Entries.constant_entry
+  | ConstantEntry of bool * private_constants Entries.constant_entry * Declarations.typing_flags
   | GlobalRecipe of Cooking.recipe
 
 type exported_private_constant = 
@@ -512,10 +512,10 @@ let add_constant dir l decl senv =
   let no_section = DirPath.is_empty dir in
   let seff_to_export, decl =
     match decl with
-    | ConstantEntry (true, ce) ->
+    | ConstantEntry (true, ce, flags) ->
         let exports, ce =
-          Term_typing.export_side_effects senv.revstruct senv.env ce in
-        exports, ConstantEntry (false, ce)
+          Term_typing.export_side_effects ~flags senv.revstruct senv.env ce in
+        exports, ConstantEntry (false, ce, flags)
     | _ -> [], decl
   in
   let senv =
@@ -524,8 +524,8 @@ let add_constant dir l decl senv =
   let senv =
     let cb = 
       match decl with
-      | ConstantEntry (export_seff,ce) ->
-        Term_typing.translate_constant senv.revstruct senv.env kn ce
+      | ConstantEntry (export_seff,ce,flags) ->
+        Term_typing.translate_constant ~flags senv.revstruct senv.env kn ce
       | GlobalRecipe r ->
         let cb = Term_typing.translate_recipe senv.env kn r in
         if no_section then Declareops.hcons_const_body cb else cb in
diff --git a/kernel/safe_typing.mli b/kernel/safe_typing.mli
index 71dac321f..b614a368c 100644
--- a/kernel/safe_typing.mli
+++ b/kernel/safe_typing.mli
@@ -77,19 +77,21 @@ val is_joined_environment : safe_environment -> bool
 (** Insertion of local declarations (Local or Variables) *)
 
 val push_named_assum :
+  flags:Declarations.typing_flags ->
   (Id.t * Term.types * bool (* polymorphic *))
     Univ.in_universe_context_set -> safe_transformer0
 
 (** Returns the full universe context necessary to typecheck the definition
   (futures are forced) *)
 val push_named_def :
+  flags:Declarations.typing_flags ->
   Id.t * private_constants Entries.definition_entry -> Univ.universe_context_set safe_transformer
 
 (** Insertion of global axioms or definitions *)
 
 type global_declaration =
                   (* bool: export private constants *)
-  | ConstantEntry of bool * private_constants Entries.constant_entry
+  | ConstantEntry of bool * private_constants Entries.constant_entry * Declarations.typing_flags
   | GlobalRecipe of Cooking.recipe
 
 type exported_private_constant = 
diff --git a/kernel/term_typing.ml b/kernel/term_typing.ml
index 6bfd2457a..8d74dc390 100644
--- a/kernel/term_typing.ml
+++ b/kernel/term_typing.ml
@@ -22,18 +22,18 @@ open Entries
 open Typeops
 open Fast_typeops
 
-let constrain_type env j poly subst = function
+let constrain_type ~flags env j poly subst = function
   | `None -> 
     if not poly then (* Old-style polymorphism *)
       make_polymorphic_if_constant_for_ind env j
     else RegularArity (Vars.subst_univs_level_constr subst j.uj_type)
   | `Some t ->
-      let tj = infer_type env t in
+      let tj = infer_type ~flags env t in
       let _ = judge_of_cast env j DEFAULTcast tj in
 	assert (eq_constr t tj.utj_val);
 	RegularArity (Vars.subst_univs_level_constr subst t)
   | `SomeWJ (t, tj) ->
-      let tj = infer_type env t in
+      let tj = infer_type ~flags env t in
       let _ = judge_of_cast env j DEFAULTcast tj in
 	assert (eq_constr t tj.utj_val);
 	RegularArity (Vars.subst_univs_level_constr subst t)
@@ -171,11 +171,11 @@ let feedback_completion_typecheck =
   Option.iter (fun state_id ->
       feedback ~id:(State state_id) Feedback.Complete)
 
-let infer_declaration ~trust env kn dcl =
+let infer_declaration ~flags ~trust env kn dcl =
   match dcl with
   | ParameterEntry (ctx,poly,(t,uctx),nl) ->
       let env = push_context ~strict:(not poly) uctx env in
-      let j = infer env t in
+      let j = infer ~flags env t in
       let abstract = poly && not (Option.is_empty kn) in
       let usubst, univs = Univ.abstract_universes abstract uctx in
       let c = Typeops.assumption_of_judgment env j in
@@ -196,7 +196,7 @@ let infer_declaration ~trust env kn dcl =
 	  let env' = push_context_set uctx env in
           let j =
             let body,env',ectx = skip_trusted_seff valid_signatures body env' in
-            let j = infer env' body in
+            let j = infer ~flags env' body in
             unzip ectx j in
           let j = hcons_j j in
 	  let subst = Univ.LMap.empty in
@@ -220,8 +220,8 @@ let infer_declaration ~trust env kn dcl =
       let abstract = c.const_entry_polymorphic && not (Option.is_empty kn) in
       let usubst, univs =
 	Univ.abstract_universes abstract (Univ.ContextSet.to_context ctx) in      
-      let j = infer env body in
-      let typ = constrain_type env j c.const_entry_polymorphic usubst (map_option_typ typ) in
+      let j = infer ~flags env body in
+      let typ = constrain_type ~flags env j c.const_entry_polymorphic usubst (map_option_typ typ) in
       let def = hcons_constr (Vars.subst_univs_level_constr usubst j.uj_val) in
       let def = 
 	if opaque then OpaqueDef (Opaqueproof.create (Future.from_val (def, Univ.ContextSet.empty)))
@@ -268,7 +268,7 @@ let record_aux env s_ty s_bo suggested_expr =
 let suggest_proof_using = ref (fun _ _ _ _ _ -> "")
 let set_suggest_proof_using f = suggest_proof_using := f
 
-let build_constant_declaration kn env (def,typ,proj,poly,univs,inline_code,ctx) =
+let build_constant_declaration ~flags kn env (def,typ,proj,poly,univs,inline_code,ctx) =
   let open Context.Named.Declaration in
   let check declared inferred =
     let mk_set l = List.fold_right Id.Set.add (List.map get_id l) Id.Set.empty in
@@ -352,7 +352,9 @@ let build_constant_declaration kn env (def,typ,proj,poly,univs,inline_code,ctx)
 	    const_body_code = None;
 	    const_polymorphic = poly;
 	    const_universes = univs;
-	    const_inline_code = inline_code }
+	    const_inline_code = inline_code;
+	    const_typing_flags = flags;
+	    }
 	in
 	let env = add_constant kn cb env in
 	compile_constant_body env comp_univs def
@@ -365,13 +367,14 @@ let build_constant_declaration kn env (def,typ,proj,poly,univs,inline_code,ctx)
     const_body_code = tps;
     const_polymorphic = poly;
     const_universes = univs;
-    const_inline_code = inline_code }
+    const_inline_code = inline_code;
+    const_typing_flags = flags }
 
 (*s Global and local constant declaration. *)
 
-let translate_constant mb env kn ce =
-  build_constant_declaration kn env
-    (infer_declaration ~trust:mb env (Some kn) ce)
+let translate_constant ~flags mb env kn ce =
+  build_constant_declaration ~flags kn env
+    (infer_declaration ~flags ~trust:mb env (Some kn) ce)
 
 let constant_entry_of_side_effect cb u =
   let pt =
@@ -406,7 +409,7 @@ type side_effect_role =
 type exported_side_effect = 
   constant * constant_body * side_effects constant_entry * side_effect_role
 
-let export_side_effects mb env ce =
+let export_side_effects ~flags mb env ce =
   match ce with
   | ParameterEntry _ | ProjectionEntry _ -> [], ce
   | DefinitionEntry c ->
@@ -447,7 +450,7 @@ let export_side_effects mb env ce =
            let env, cbs =
              List.fold_left (fun (env,cbs) (kn, ocb, u, r) ->
                let ce = constant_entry_of_side_effect ocb u in
-               let cb = translate_constant mb env kn ce in
+               let cb = translate_constant ~flags mb env kn ce in
                (push_seff env (kn, cb,`Nothing, Subproof),(kn,cb,ce,r) :: cbs)) 
              (env,[]) cbs in
            translate_seff sl rest (cbs @ acc) env
@@ -462,17 +465,17 @@ let export_side_effects mb env ce =
        translate_seff trusted seff [] env
 ;;
 
-let translate_local_assum env t =
-  let j = infer env t in
+let translate_local_assum ~flags env t =
+  let j = infer ~flags env t in
   let t = Typeops.assumption_of_judgment env j in
     t
 
 let translate_recipe env kn r =
-  build_constant_declaration kn env (Cooking.cook_constant env r)
+  build_constant_declaration ~flags:{check_guarded=true} kn env (Cooking.cook_constant env r)
 
-let translate_local_def mb env id centry =
+let translate_local_def ~flags mb env id centry =
   let def,typ,proj,poly,univs,inline_code,ctx =
-    infer_declaration ~trust:mb env None (DefinitionEntry centry) in
+    infer_declaration ~flags ~trust:mb env None (DefinitionEntry centry) in
   let typ = type_of_constant_type env typ in
   if ctx = None && !Flags.compilation_mode = Flags.BuildVo then begin
     match def with
diff --git a/kernel/term_typing.mli b/kernel/term_typing.mli
index fcd95576c..b635f2d31 100644
--- a/kernel/term_typing.mli
+++ b/kernel/term_typing.mli
@@ -12,10 +12,10 @@ open Environ
 open Declarations
 open Entries
 
-val translate_local_def : structure_body -> env -> Id.t -> side_effects definition_entry ->
+val translate_local_def : flags:typing_flags -> structure_body -> env -> Id.t -> side_effects definition_entry ->
   constant_def * types * constant_universes
 
-val translate_local_assum : env -> types -> types
+val translate_local_assum : flags:typing_flags -> env -> types -> types
 
 val mk_pure_proof : constr -> side_effects proof_output
 
@@ -32,7 +32,7 @@ val inline_entry_side_effects :
 val uniq_seff : side_effects -> side_effects
 
 val translate_constant :
-  structure_body -> env -> constant -> side_effects constant_entry ->
+  flags:typing_flags -> structure_body -> env -> constant -> side_effects constant_entry ->
     constant_body
 
 type side_effect_role =
@@ -47,7 +47,7 @@ type exported_side_effect =
  * be pushed in the safe_env by safe typing.  The main constant entry
  * needs to be translated as usual after this step. *)
 val export_side_effects :
-  structure_body -> env -> side_effects constant_entry ->
+  flags:typing_flags -> structure_body -> env -> side_effects constant_entry ->
     exported_side_effect list * side_effects constant_entry
 
 val constant_entry_of_side_effect :
@@ -60,11 +60,11 @@ val translate_recipe : env -> constant -> Cooking.recipe -> constant_body
 
 (** Internal functions, mentioned here for debug purpose only *)
 
-val infer_declaration : trust:structure_body -> env -> constant option -> 
+val infer_declaration : flags:typing_flags -> trust:structure_body -> env -> constant option -> 
   side_effects constant_entry -> Cooking.result
 
 val build_constant_declaration :
-  constant -> env -> Cooking.result -> constant_body
+  flags:typing_flags -> constant -> env -> Cooking.result -> constant_body
 
 val set_suggest_proof_using :
   (string -> env -> Id.Set.t -> Id.Set.t -> Id.t list -> string) -> unit
diff --git a/kernel/typeops.ml b/kernel/typeops.ml
index 0ea68e2bc..a94a049df 100644
--- a/kernel/typeops.ml
+++ b/kernel/typeops.ml
@@ -500,13 +500,13 @@ let rec execute env cstr =
     | Fix ((vn,i as vni),recdef) ->
       let (fix_ty,recdef') = execute_recdef env recdef i in
       let fix = (vni,recdef') in
-        check_fix env fix;
+        check_fix ~flags:{check_guarded=true} env fix;
 	make_judge (mkFix fix) fix_ty
 	  
     | CoFix (i,recdef) ->
       let (fix_ty,recdef') = execute_recdef env recdef i in
       let cofix = (i,recdef') in
-        check_cofix env cofix;
+        check_cofix ~flags:{check_guarded=true} env cofix;
 	(make_judge (mkCoFix cofix) fix_ty)
 	  
     (* Partial proofs: unsupported by the kernel *)
diff --git a/library/declare.ml b/library/declare.ml
index 84284fd18..9ec299bed 100644
--- a/library/declare.ml
+++ b/library/declare.ml
@@ -58,11 +58,11 @@ let cache_variable ((sp,_),o) =
 
   let impl,opaq,poly,ctx = match d with (* Fails if not well-typed *)
     | SectionLocalAssum ((ty,ctx),poly,impl) ->
-      let () = Global.push_named_assum ((id,ty,poly),ctx) in
+      let () = Global.push_named_assum ~flags:{check_guarded=true} ((id,ty,poly),ctx) in
       let impl = if impl then Implicit else Explicit in
 	impl, true, poly, ctx
     | SectionLocalDef (de) ->
-      let univs = Global.push_named_def (id,de) in
+      let univs = Global.push_named_def ~flags:{check_guarded=true} (id,de) in
       Explicit, de.const_entry_opaque,
       de.const_entry_polymorphic, univs in
   Nametab.push (Nametab.Until 1) (restrict_path 0 sp) (VarRef id);
@@ -180,7 +180,7 @@ let discharge_constant ((sp, kn), obj) =
 (* Hack to reduce the size of .vo: we keep only what load/open needs *)
 let dummy_constant_entry = 
   ConstantEntry
-    (false, ParameterEntry (None,false,(mkProp,Univ.UContext.empty),None))
+    (false, ParameterEntry (None,false,(mkProp,Univ.UContext.empty),None), {check_guarded=true})
 
 let dummy_constant cst = {
   cst_decl = dummy_constant_entry;
@@ -205,7 +205,7 @@ let (inConstant, outConstant : (constant_obj -> obj) * (obj -> constant_obj)) =
 let declare_scheme = ref (fun _ _ -> assert false)
 let set_declare_scheme f = declare_scheme := f
 
-let declare_constant_common id cst =
+let declare_constant_common ~flags id cst =
   let update_tables c =
 (*  Printf.eprintf "tables: %s\n%!" (Names.Constant.to_string c); *)
     declare_constant_implicits c;
@@ -216,7 +216,7 @@ let declare_constant_common id cst =
   List.iter (fun (c,ce,role) ->
       (* handling of private_constants just exported *)
       let o = inConstant {
-        cst_decl = ConstantEntry (false, ce);
+        cst_decl = ConstantEntry (false, ce, flags);
         cst_hyps = [] ;
         cst_kind = IsProof Theorem;
         cst_locl = false;
@@ -246,7 +246,7 @@ let definition_entry ?fix_exn ?(opaque=false) ?(inline=false) ?types
     const_entry_feedback = None;
     const_entry_inline_code = inline}
 
-let declare_constant ?(internal = UserIndividualRequest) ?(local = false) id ?(export_seff=false) (cd, kind) =
+let declare_constant ?(flags={check_guarded=true}) ?(internal = UserIndividualRequest) ?(local = false) id ?(export_seff=false) (cd, kind) =
   let export = (* We deal with side effects *)
     match cd with
     | DefinitionEntry de when
@@ -259,24 +259,24 @@ let declare_constant ?(internal = UserIndividualRequest) ?(local = false) id ?(e
     | _ -> false
   in
   let cst = {
-    cst_decl = ConstantEntry (export,cd);
+    cst_decl = ConstantEntry (export,cd,flags);
     cst_hyps = [] ;
     cst_kind = kind;
     cst_locl = local;
     cst_exported = [];
     cst_was_seff = false;
   } in
-  let kn = declare_constant_common id cst in
+  let kn = declare_constant_common id cst ~flags in
   let () = if_xml (Hook.get f_xml_declare_constant) (internal, kn) in
   kn
 
-let declare_definition ?(internal=UserIndividualRequest)
+let declare_definition ?flags ?(internal=UserIndividualRequest)
   ?(opaque=false) ?(kind=Decl_kinds.Definition) ?(local = false)
   ?(poly=false) id ?types (body,ctx) =
   let cb =
     definition_entry ?types ~poly ~univs:(Univ.ContextSet.to_context ctx) ~opaque body
   in
-    declare_constant ~internal ~local id
+    declare_constant ?flags ~internal ~local id
       (Entries.DefinitionEntry cb, Decl_kinds.IsDefinition kind)
 
 (** Declaration of inductive blocks *)
@@ -353,7 +353,8 @@ let dummy_inductive_entry (_,m) = ([],{
   mind_entry_inds = List.map dummy_one_inductive_entry m.mind_entry_inds;
   mind_entry_polymorphic = false;
   mind_entry_universes = Univ.UContext.empty;
-  mind_entry_private = None })
+  mind_entry_private = None;
+  mind_entry_check_positivity = true; })
 
 type inductive_obj = Dischargedhypsmap.discharged_hyps * mutual_inductive_entry
 
@@ -373,7 +374,7 @@ let declare_projections mind =
       Array.iteri (fun i kn ->
 	let id = Label.to_id (Constant.label kn) in
 	let entry = {proj_entry_ind = mind; proj_entry_arg = i} in
-	let kn' = declare_constant id (ProjectionEntry entry,
+	let kn' = declare_constant ~flags:{check_guarded=true} id (ProjectionEntry entry,
 				       IsDefinition StructureComponent)
 	in
 	  assert(eq_constant kn kn')) kns; true,true
diff --git a/library/declare.mli b/library/declare.mli
index 8dd24d278..3ba63b5a6 100644
--- a/library/declare.mli
+++ b/library/declare.mli
@@ -54,9 +54,11 @@ val definition_entry : ?fix_exn:Future.fix_exn ->
   ?eff:Safe_typing.private_constants -> constr -> Safe_typing.private_constants definition_entry
 
 val declare_constant :
+  ?flags:Declarations.typing_flags -> (** default [check_guarded=true] *)
  ?internal:internal_flag -> ?local:bool -> Id.t -> ?export_seff:bool -> constant_declaration -> constant
 
 val declare_definition : 
+  ?flags:Declarations.typing_flags -> (** default [check_guarded=true] *)
   ?internal:internal_flag -> ?opaque:bool -> ?kind:definition_object_kind ->
   ?local:bool -> ?poly:polymorphic -> Id.t -> ?types:constr -> 
   constr Univ.in_universe_context_set -> constant
diff --git a/library/global.ml b/library/global.ml
index 2398e92b0..f4ee62b6e 100644
--- a/library/global.ml
+++ b/library/global.ml
@@ -77,8 +77,8 @@ let globalize_with_summary fs f =
 
 let i2l = Label.of_id
 
-let push_named_assum a = globalize0 (Safe_typing.push_named_assum a)
-let push_named_def d = globalize (Safe_typing.push_named_def d)
+let push_named_assum ~flags a = globalize0 (Safe_typing.push_named_assum ~flags a)
+let push_named_def ~flags d = globalize (Safe_typing.push_named_def ~flags d)
 let add_constraints c = globalize0 (Safe_typing.add_constraints c)
 let push_context_set b c = globalize0 (Safe_typing.push_context_set b c)
 let push_context b c = globalize0 (Safe_typing.push_context b c)
diff --git a/library/global.mli b/library/global.mli
index bf653307c..7c6cecb4e 100644
--- a/library/global.mli
+++ b/library/global.mli
@@ -30,8 +30,8 @@ val set_engagement : Declarations.engagement -> unit
 
 (** Variables, Local definitions, constants, inductive types *)
 
-val push_named_assum : (Id.t * Constr.types * bool) Univ.in_universe_context_set -> unit
-val push_named_def   : (Id.t * Safe_typing.private_constants Entries.definition_entry) -> Univ.universe_context_set
+val push_named_assum : flags:Declarations.typing_flags -> (Id.t * Constr.types * bool) Univ.in_universe_context_set -> unit
+val push_named_def   : flags:Declarations.typing_flags -> (Id.t * Safe_typing.private_constants Entries.definition_entry) -> Univ.universe_context_set
 
 val add_constant :
   DirPath.t -> Id.t -> Safe_typing.global_declaration ->
diff --git a/plugins/funind/glob_term_to_relation.ml b/plugins/funind/glob_term_to_relation.ml
index c424fe122..f69c4d821 100644
--- a/plugins/funind/glob_term_to_relation.ml
+++ b/plugins/funind/glob_term_to_relation.ml
@@ -1432,7 +1432,7 @@ let do_build_inductive
 (*   in *)
   let _time2 = System.get_time () in
   try
-    with_full_print (Flags.silently (Command.do_mutual_inductive rel_inds (Flags.is_universe_polymorphism ()) false)) Decl_kinds.Finite
+    with_full_print (Flags.silently (Command.do_mutual_inductive true rel_inds (Flags.is_universe_polymorphism ()) false)) Decl_kinds.Finite
   with
     | UserError(s,msg) as e ->
 	let _time3 = System.get_time () in
diff --git a/plugins/funind/indfun.ml b/plugins/funind/indfun.ml
index 0cacb003d..d8340dddb 100644
--- a/plugins/funind/indfun.ml
+++ b/plugins/funind/indfun.ml
@@ -400,7 +400,7 @@ let register_struct is_rec (fixpoint_exprl:(Vernacexpr.fixpoint_expr * Vernacexp
        in
        evd,List.rev rev_pconstants
     | _ ->
-       Command.do_fixpoint Global (Flags.is_universe_polymorphism ()) fixpoint_exprl;
+       Command.do_fixpoint ~flags:{Declarations.check_guarded=true} Global (Flags.is_universe_polymorphism ()) fixpoint_exprl;
        let evd,rev_pconstants =
 	 List.fold_left
 	   (fun (evd,l) ((((_,fname),_),_,_,_,_),_) ->
diff --git a/plugins/funind/merge.ml b/plugins/funind/merge.ml
index 99a165044..a78eb1af7 100644
--- a/plugins/funind/merge.ml
+++ b/plugins/funind/merge.ml
@@ -887,7 +887,7 @@ let merge_inductive (ind1: inductive) (ind2: inductive)
   let indexpr = glob_constr_list_to_inductive_expr prms1 prms2 mib1 mib2 shift_prm rawlist in
   (* Declare inductive *)
   let indl,_,_ = Command.extract_mutual_inductive_declaration_components [(indexpr,[])] in
-  let mie,pl,impls = Command.interp_mutual_inductive indl []
+  let mie,pl,impls = Command.interp_mutual_inductive true indl []
           false (*FIXMEnon-poly *) false (* means not private *) Decl_kinds.Finite (* means: not coinductive *) in
   (* Declare the mutual inductive block with its associated schemes *)
   ignore (Command.declare_mutual_inductive_with_eliminations mie pl impls)
diff --git a/pretyping/inductiveops.ml b/pretyping/inductiveops.ml
index 80f1988a9..0b488308a 100644
--- a/pretyping/inductiveops.ml
+++ b/pretyping/inductiveops.ml
@@ -592,9 +592,9 @@ let type_of_projection_knowing_arg env sigma p c ty =
 let control_only_guard env c =
   let check_fix_cofix e c = match kind_of_term c with
     | CoFix (_,(_,_,_) as cofix) ->
-	Inductive.check_cofix e cofix
+      Inductive.check_cofix ~flags:{check_guarded=true} e cofix
     | Fix (_,(_,_,_) as fix) ->
-	Inductive.check_fix e fix
+      Inductive.check_fix ~flags:{check_guarded=true} e fix
     | _ -> ()
   in
   let rec iter env c =
diff --git a/pretyping/pretyping.ml b/pretyping/pretyping.ml
index 378294c98..c894d96a7 100644
--- a/pretyping/pretyping.ml
+++ b/pretyping/pretyping.ml
@@ -72,14 +72,17 @@ open Inductiveops
 
 exception Found of int array
 
-let search_guard loc env possible_indexes fixdefs =
+(* spiwack: I chose [tflags] rather than [flags], like in the rest of
+   the code, for the argument name to avoid interference with the
+   argument for [inference_flags] also used in this module. *)
+let search_guard ~tflags loc env possible_indexes fixdefs =
   (* Standard situation with only one possibility for each fix. *)
   (* We treat it separately in order to get proper error msg. *)
   let is_singleton = function [_] -> true | _ -> false in
   if List.for_all is_singleton possible_indexes then
     let indexes = Array.of_list (List.map List.hd possible_indexes) in
     let fix = ((indexes, 0),fixdefs) in
-    (try check_fix env fix
+    (try check_fix env ~flags:tflags fix
      with reraise ->
        let (e, info) = Errors.push reraise in
        let info = Loc.add_loc info loc in
@@ -91,8 +94,14 @@ let search_guard loc env possible_indexes fixdefs =
        List.iter
 	 (fun l ->
 	    let indexes = Array.of_list l in
-	    let fix = ((indexes, 0),fixdefs) in
-	    try check_fix env fix; raise (Found indexes)
+            let fix = ((indexes, 0),fixdefs) in
+            (* spiwack: We search for a unspecified structural
+               argument under the assumption that we need to check the
+               guardedness condition (otherwise the first inductive argument
+               will be chosen). A more robust solution may be to raise an
+               error when totality is assumed but the strutural argument is
+               not specified. *)
+            try check_fix env ~flags:{Declarations.check_guarded=true} fix; raise (Found indexes)
 	    with TypeError _ -> ())
 	 (List.combinations possible_indexes);
        let errmsg = "Cannot guess decreasing argument of fix." in
@@ -606,11 +615,15 @@ let rec pretype k0 resolve_tc (tycon : type_constraint) env evdref (lvar : ltac_
 			     vn)
 	  in
 	  let fixdecls = (names,ftys,fdefs) in
-	  let indexes = search_guard loc env possible_indexes fixdecls in
+          let indexes =
+            search_guard
+              ~tflags:{Declarations.check_guarded=true}
+              loc env possible_indexes fixdecls
+          in
 	    make_judge (mkFix ((indexes,i),fixdecls)) ftys.(i)
 	| GCoFix i ->
 	  let cofix = (i,(names,ftys,fdefs)) in
-	    (try check_cofix env cofix
+            (try check_cofix env ~flags:{Declarations.check_guarded=true} cofix
              with reraise ->
                let (e, info) = Errors.push reraise in
                let info = Loc.add_loc info loc in
diff --git a/pretyping/pretyping.mli b/pretyping/pretyping.mli
index 824bb11aa..2c02b4a21 100644
--- a/pretyping/pretyping.mli
+++ b/pretyping/pretyping.mli
@@ -22,7 +22,7 @@ open Misctypes
 
 (** An auxiliary function for searching for fixpoint guard indexes *)
 
-val search_guard :
+val search_guard : tflags:Declarations.typing_flags ->
   Loc.t -> env -> int list list -> rec_declaration -> int array
 
 type typing_constraint = OfType of types | IsType | WithoutTypeConstraint
diff --git a/pretyping/typing.ml b/pretyping/typing.ml
index 52afa7f83..598dd16d0 100644
--- a/pretyping/typing.ml
+++ b/pretyping/typing.ml
@@ -189,13 +189,13 @@ let rec execute env evdref cstr =
     | Fix ((vn,i as vni),recdef) ->
         let (_,tys,_ as recdef') = execute_recdef env evdref recdef in
 	let fix = (vni,recdef') in
-        check_fix env fix;
+        check_fix env ~flags:{Declarations.check_guarded=true} fix;
 	make_judge (mkFix fix) tys.(i)
 
     | CoFix (i,recdef) ->
         let (_,tys,_ as recdef') = execute_recdef env evdref recdef in
         let cofix = (i,recdef') in
-        check_cofix env cofix;
+        check_cofix env ~flags:{Declarations.check_guarded=true} cofix;
 	make_judge (mkCoFix cofix) tys.(i)
 
     | Sort (Prop c) ->
diff --git a/printing/printer.ml b/printing/printer.ml
index cc8da4097..2357ca0ea 100644
--- a/printing/printer.ml
+++ b/printing/printer.ml
@@ -697,9 +697,14 @@ let prterm = pr_lconstr
 (* Printer function for sets of Assumptions.assumptions.
    It is used primarily by the Print Assumptions command. *)
 
+type axiom =
+  | Constant of constant (* An axiom or a constant. *)
+  | Positive of MutInd.t (* A mutually inductive definition which has been assumed positive. *)
+  | Guarded of constant (* a constant whose (co)fixpoints have been assumed to be guarded *)
+
 type context_object =
   | Variable of Id.t (* A section variable or a Let definition *)
-  | Axiom of constant * (Label.t * Context.Rel.t * types) list
+  | Axiom of axiom * (Label.t * Context.Rel.t * types) list
   | Opaque of constant     (* An opaque constant. *)
   | Transparent of constant
 
@@ -707,12 +712,25 @@ type context_object =
 module OrderedContextObject =
 struct
   type t = context_object
+
+  let compare_axiom x y =
+    match x,y with
+    | Constant k1 , Constant k2 ->
+        con_ord k1 k2
+    | Positive m1 , Positive m2 ->
+        MutInd.CanOrd.compare m1 m2
+    | Guarded k1 , Guarded k2 ->
+        con_ord k1 k2
+    | _ , Constant _ -> 1
+    | _ , Positive _ -> 1
+    | _ -> -1
+
   let compare x y =
     match x , y with
     | Variable i1 , Variable i2 -> Id.compare i1 i2
     | Variable _ , _ -> -1
     | _ , Variable _ -> 1
-    | Axiom (k1,_) , Axiom (k2, _) -> con_ord k1 k2
+    | Axiom (k1,_) , Axiom (k2, _) -> compare_axiom k1 k2
     | Axiom _ , _ -> -1
     | _ , Axiom _ -> 1
     | Opaque k1 , Opaque k2 -> con_ord k1 k2
@@ -745,17 +763,26 @@ let pr_assumptionset env s =
       try str " " ++ pr_ltype_env env sigma typ
       with e when Errors.noncritical e -> mt ()
     in
+    let pr_axiom env ax typ =
+      match ax with
+      | Constant kn ->
+          safe_pr_constant env kn ++ safe_pr_ltype typ
+      | Positive m ->
+          hov 2 (MutInd.print m ++ spc () ++ strbrk"is positive.")
+      | Guarded kn ->
+          hov 2 (safe_pr_constant env kn ++ spc () ++ strbrk"is positive.")
+    in
     let fold t typ accu =
       let (v, a, o, tr) = accu in
       match t with
       | Variable id ->
         let var = pr_id id ++ str " : " ++ pr_ltype typ in
         (var :: v, a, o, tr)
-      | Axiom (kn,[]) ->
-        let ax = safe_pr_constant env kn ++ safe_pr_ltype typ in
+      | Axiom (axiom, []) ->
+        let ax = pr_axiom env axiom typ in
         (v, ax :: a, o, tr)
-      | Axiom (kn,l) ->
-        let ax = safe_pr_constant env kn ++ safe_pr_ltype typ ++
+      | Axiom (axiom,l) ->
+        let ax = pr_axiom env axiom typ ++
           cut() ++
           prlist_with_sep cut (fun (lbl, ctx, ty) ->
             str " used in " ++ pr_label lbl ++
diff --git a/printing/printer.mli b/printing/printer.mli
index 70993bb72..695ab33b2 100644
--- a/printing/printer.mli
+++ b/printing/printer.mli
@@ -161,12 +161,16 @@ val prterm                 : constr -> std_ppcmds (** = pr_lconstr *)
 
 
 (** Declarations for the "Print Assumption" command *)
+type axiom =
+  | Constant of constant (* An axiom or a constant. *)
+  | Positive of MutInd.t (* A mutually inductive definition which has been assumed positive. *)
+  | Guarded of constant (* a constant whose (co)fixpoints have been assumed to be guarded *)
+
 type context_object =
-  | Variable of Id.t  (** A section variable or a Let definition *)
-  (** An axiom and the type it inhabits (if an axiom of the empty type) *)
-  | Axiom of constant * (Label.t * Context.Rel.t * types) list
-  | Opaque of constant      (** An opaque constant. *)
-  | Transparent of constant (** A transparent constant *)
+  | Variable of Id.t (* A section variable or a Let definition *)
+  | Axiom of axiom * (Label.t * Context.Rel.t * types) list
+  | Opaque of constant     (* An opaque constant. *)
+  | Transparent of constant
 
 module ContextObjectSet : Set.S with type elt = context_object
 module ContextObjectMap : CMap.ExtS
diff --git a/stm/lemmas.ml b/stm/lemmas.ml
index 0a63a3a0f..b84b1265e 100644
--- a/stm/lemmas.ml
+++ b/stm/lemmas.ml
@@ -79,7 +79,7 @@ let adjust_guardness_conditions const = function
                     List.fold_left (fun e (_,c,cb,_) -> add c cb e) env l)
                 env (Safe_typing.side_effects_of_private_constants eff) in
               let indexes =
-	        search_guard Loc.ghost env
+                search_guard ~tflags:{Declarations.check_guarded=true} Loc.ghost env
                   possible_indexes fixdecls in
 		(mkFix ((indexes,0),fixdecls), ctx), eff
           | _ -> (body, ctx), eff) }
diff --git a/toplevel/assumptions.ml b/toplevel/assumptions.ml
index 1802b2d36..c05c5f6c2 100644
--- a/toplevel/assumptions.ml
+++ b/toplevel/assumptions.ml
@@ -286,11 +286,17 @@ let assumptions ?(add_opaque=false) ?(add_transparent=false) st gr t =
     if is_local_assum decl then ContextObjectMap.add (Variable id) t accu
     else accu
   | ConstRef kn ->
-    let cb = lookup_constant kn in
+      let cb = lookup_constant kn in
+      let accu =
+        if cb.const_typing_flags.check_guarded then accu
+        else
+          let l = try Refmap_env.find obj ax2ty with Not_found -> [] in
+          ContextObjectMap.add (Axiom (Guarded kn, l)) Constr.mkProp accu
+      in
     if not (Declareops.constant_has_body cb) then
       let t = type_of_constant cb in
       let l = try Refmap_env.find obj ax2ty with Not_found -> [] in
-      ContextObjectMap.add (Axiom (kn,l)) t accu
+      ContextObjectMap.add (Axiom (Constant kn,l)) t accu
     else if add_opaque && (Declareops.is_opaque cb || not (Cpred.mem kn knst)) then
       let t = type_of_constant cb in
       ContextObjectMap.add (Opaque kn) t accu
@@ -299,6 +305,12 @@ let assumptions ?(add_opaque=false) ?(add_transparent=false) st gr t =
       ContextObjectMap.add (Transparent kn) t accu
     else
       accu
-  | IndRef _ | ConstructRef _ -> accu
+  | IndRef (m,_) | ConstructRef ((m,_),_) ->
+      let mind = Global.lookup_mind m in
+      if mind.mind_checked_positive then
+        accu
+      else
+        let l = try Refmap_env.find obj ax2ty with Not_found -> [] in
+        ContextObjectMap.add (Axiom (Positive m, l)) Constr.mkProp accu
   in
   Refmap_env.fold fold graph ContextObjectMap.empty
diff --git a/toplevel/command.ml b/toplevel/command.ml
index f0f678e08..6f2dd1bf1 100644
--- a/toplevel/command.ml
+++ b/toplevel/command.ml
@@ -145,9 +145,9 @@ let get_locality id = function
 | Local -> true
 | Global -> false
 
-let declare_global_definition ident ce local k pl imps =
+let declare_global_definition ~flags ident ce local k pl imps =
   let local = get_locality ident local in
-  let kn = declare_constant ident ~local (DefinitionEntry ce, IsDefinition k) in
+  let kn = declare_constant ~flags ident ~local (DefinitionEntry ce, IsDefinition k) in
   let gr = ConstRef kn in
   let () = maybe_declare_manual_implicits false gr imps in
   let () = Universes.register_universe_binders gr pl in
@@ -158,7 +158,7 @@ let declare_definition_hook = ref ignore
 let set_declare_definition_hook = (:=) declare_definition_hook
 let get_declare_definition_hook () = !declare_definition_hook
 
-let declare_definition ident (local, p, k) ce pl imps hook =
+let declare_definition ~flags ident (local, p, k) ce pl imps hook =
   let fix_exn = Future.fix_exn_of ce.const_entry_body in
   let () = !declare_definition_hook ce in
   let r = match local with
@@ -175,11 +175,11 @@ let declare_definition ident (local, p, k) ce pl imps hook =
     in
     gr
   | Discharge | Local | Global ->
-    declare_global_definition ident ce local k pl imps in
+    declare_global_definition ~flags ident ce local k pl imps in
   Lemmas.call_hook fix_exn hook local r
 
 let _ = Obligations.declare_definition_ref :=
-       (fun i k c imps hook -> declare_definition i k c [] imps hook)
+       (fun i k c imps hook -> declare_definition ~flags:{Declarations.check_guarded=true} i k c [] imps hook)
 
 let do_definition ident k pl bl red_option c ctypopt hook =
   let (ce, evd, pl', imps as def) =
@@ -203,7 +203,7 @@ let do_definition ident k pl bl red_option c ctypopt hook =
 	ignore(Obligations.add_definition
           ident ~term:c cty ctx ?pl ~implicits:imps ~kind:k ~hook obls)
     else let ce = check_definition def in
-      ignore(declare_definition ident k ce pl' imps
+      ignore(declare_definition ~flags:{Declarations.check_guarded=true} ident k ce pl' imps
         (Lemmas.mk_hook
           (fun l r -> Lemmas.call_hook (fun exn -> exn) hook l r;r)))
 
@@ -549,7 +549,7 @@ let check_param = function
 | LocalRawAssum (nas, Default _, _) -> List.iter check_named nas
 | LocalRawAssum (nas, Generalized _, _) -> ()
 
-let interp_mutual_inductive (paramsl,indl) notations poly prv finite =
+let interp_mutual_inductive chk (paramsl,indl) notations poly prv finite =
   check_all_names_different indl;
   List.iter check_param paramsl;
   let env0 = Global.env() in
@@ -630,7 +630,8 @@ let interp_mutual_inductive (paramsl,indl) notations poly prv finite =
     mind_entry_inds = entries;
     mind_entry_polymorphic = poly;
     mind_entry_private = if prv then Some false else None;
-    mind_entry_universes = uctx },
+    mind_entry_universes = uctx;
+    mind_entry_check_positivity = chk; },
     pl, impls
 
 (* Very syntactical equality *)
@@ -715,17 +716,20 @@ type one_inductive_impls =
   Impargs.manual_explicitation list (* for inds *)*
   Impargs.manual_explicitation list list (* for constrs *)
 
-let do_mutual_inductive indl poly prv finite =
+let do_mutual_inductive chk indl poly prv finite =
   let indl,coes,ntns = extract_mutual_inductive_declaration_components indl in
   (* Interpret the types *)
-  let mie,pl,impls = interp_mutual_inductive indl ntns poly prv finite in
+  let mie,pl,impls = interp_mutual_inductive chk indl ntns poly prv finite in
   (* Declare the mutual inductive block with its associated schemes *)
   ignore (declare_mutual_inductive_with_eliminations mie pl impls);
   (* Declare the possible notations of inductive types *)
   List.iter Metasyntax.add_notation_interpretation ntns;
   (* Declare the coercions *)
-  List.iter (fun qid -> Class.try_add_new_coercion (locate qid) false poly) coes
-	    
+  List.iter (fun qid -> Class.try_add_new_coercion (locate qid) false poly) coes;
+  (* If [chk] is [false] (i.e. positivity is assumed) declares itself
+     as unsafe. *)
+  if not chk then Feedback.feedback Feedback.AddedAxiom else ()
+
 (* 3c| Fixpoints and co-fixpoints *)
 
 (* An (unoptimized) function that maps preorders to partial orders...
@@ -829,12 +833,12 @@ let interp_fix_body env_rec evdref impls (_,ctx) fix ccl =
 
 let build_fix_type (_,ctx) ccl = it_mkProd_or_LetIn ccl ctx
 
-let declare_fix ?(opaque = false) (_,poly,_ as kind) pl ctx f ((def,_),eff) t imps =
+let declare_fix ~flags ?(opaque = false) (_,poly,_ as kind) pl ctx f ((def,_),eff) t imps =
   let ce = definition_entry ~opaque ~types:t ~poly ~univs:ctx ~eff def in
-  declare_definition f kind ce pl imps (Lemmas.mk_hook (fun _ r -> r))
+  declare_definition ~flags f kind ce pl imps (Lemmas.mk_hook (fun _ r -> r))
 
 let _ = Obligations.declare_fix_ref :=
-	  (fun ?opaque k ctx f d t imps -> declare_fix ?opaque k [] ctx f d t imps)
+  (fun ?opaque k ctx f d t imps -> declare_fix ~flags:{Declarations.check_guarded=true} ?opaque k [] ctx f d t imps)
 
 let prepare_recursive_declaration fixnames fixtypes fixdefs =
   let defs = List.map (subst_vars (List.rev fixnames)) fixdefs in
@@ -1135,7 +1139,7 @@ let interp_cofixpoint l ntns =
   check_recursive false  env evd fix;
   (fix,pl,Evd.evar_universe_context evd,info)
     
-let declare_fixpoint local poly ((fixnames,fixdefs,fixtypes),pl,ctx,fiximps) indexes ntns =
+let declare_fixpoint ~flags local poly ((fixnames,fixdefs,fixtypes),pl,ctx,fiximps) indexes ntns =
   if List.exists Option.is_empty fixdefs then
     (* Some bodies to define by proof *)
     let thms =
@@ -1155,7 +1159,7 @@ let declare_fixpoint local poly ((fixnames,fixdefs,fixtypes),pl,ctx,fiximps) ind
     let fixdefs = List.map Option.get fixdefs in
     let fixdecls = prepare_recursive_declaration fixnames fixtypes fixdefs in
     let env = Global.env() in
-    let indexes = search_guard Loc.ghost env indexes fixdecls in
+    let indexes = search_guard ~tflags:flags Loc.ghost env indexes fixdecls in
     let fiximps = List.map (fun (n,r,p) -> r) fiximps in
     let vars = Universes.universes_of_constr (mkFix ((indexes,0),fixdecls)) in
     let fixdecls =
@@ -1164,7 +1168,7 @@ let declare_fixpoint local poly ((fixnames,fixdefs,fixtypes),pl,ctx,fiximps) ind
     let evd = Evd.restrict_universe_context evd vars in
     let fixdecls = List.map Safe_typing.mk_pure_proof fixdecls in
     let pl, ctx = Evd.universe_context ?names:pl evd in
-    ignore (List.map4 (declare_fix (local, poly, Fixpoint) pl ctx)
+    ignore (List.map4 (declare_fix ~flags (local, poly, Fixpoint) pl ctx)
 	      fixnames fixdecls fixtypes fiximps);
     (* Declare the recursive definitions *)
     fixpoint_message (Some indexes) fixnames;
@@ -1172,7 +1176,7 @@ let declare_fixpoint local poly ((fixnames,fixdefs,fixtypes),pl,ctx,fiximps) ind
   (* Declare notations *)
   List.iter Metasyntax.add_notation_interpretation ntns
 
-let declare_cofixpoint local poly ((fixnames,fixdefs,fixtypes),pl,ctx,fiximps) ntns =
+let declare_cofixpoint ~flags local poly ((fixnames,fixdefs,fixtypes),pl,ctx,fiximps) ntns =
   if List.exists Option.is_empty fixdefs then
     (* Some bodies to define by proof *)
     let thms =
@@ -1198,7 +1202,7 @@ let declare_cofixpoint local poly ((fixnames,fixdefs,fixtypes),pl,ctx,fiximps) n
     let evd = Evd.from_ctx ctx in
     let evd = Evd.restrict_universe_context evd vars in
     let pl, ctx = Evd.universe_context ?names:pl evd in
-    ignore (List.map4 (declare_fix (local, poly, CoFixpoint) pl ctx) 
+    ignore (List.map4 (declare_fix ~flags (local, poly, CoFixpoint) pl ctx) 
 	      fixnames fixdecls fixtypes fiximps);
     (* Declare the recursive definitions *)
     cofixpoint_message fixnames
@@ -1264,8 +1268,14 @@ let do_program_recursive local p fixkind fixl ntns =
 	Array.of_list (List.map (subst_vars (List.rev fixnames)) fixdefs)
       in
       let indexes = 
-	Pretyping.search_guard Loc.ghost (Global.env ()) possible_indexes fixdecls in
-	List.iteri (fun i _ -> Inductive.check_fix env ((indexes,i),fixdecls)) fixl
+        Pretyping.search_guard
+          ~tflags:{Declarations.check_guarded=true}
+          Loc.ghost (Global.env ()) possible_indexes fixdecls in
+      List.iteri (fun i _ ->
+          Inductive.check_fix env
+                              ~flags:{Declarations.check_guarded=true}
+                              ((indexes,i),fixdecls))
+        fixl
   end in
   let ctx = Evd.evar_universe_context evd in
   let kind = match fixkind with
@@ -1299,19 +1309,21 @@ let do_program_fixpoint local poly l =
 	errorlabstrm "do_program_fixpoint"
 	  (str "Well-founded fixpoints not allowed in mutually recursive blocks")
 
-let do_fixpoint local poly l =
+let do_fixpoint ~flags local poly l =
   if Flags.is_program_mode () then do_program_fixpoint local poly l
   else
     let fixl, ntns = extract_fixpoint_components true l in
     let (_, _, _, info as fix) = interp_fixpoint fixl ntns in
     let possible_indexes =
       List.map compute_possible_guardness_evidences info in
-    declare_fixpoint local poly fix possible_indexes ntns
+    declare_fixpoint ~flags local poly fix possible_indexes ntns;
+    if not flags.Declarations.check_guarded then Feedback.feedback Feedback.AddedAxiom else ()
 
-let do_cofixpoint local poly l =
+let do_cofixpoint ~flags local poly l =
   let fixl,ntns = extract_cofixpoint_components l in
     if Flags.is_program_mode () then
       do_program_recursive local poly Obligations.IsCoFixpoint fixl ntns
     else
       let cofix = interp_cofixpoint fixl ntns in
-	declare_cofixpoint local poly cofix ntns
+      declare_cofixpoint ~flags local poly cofix ntns;
+      if not flags.Declarations.check_guarded then Feedback.feedback Feedback.AddedAxiom else ()
diff --git a/toplevel/command.mli b/toplevel/command.mli
index b97cb487d..2d27552a1 100644
--- a/toplevel/command.mli
+++ b/toplevel/command.mli
@@ -36,7 +36,7 @@ val interp_definition :
   constr_expr option -> Safe_typing.private_constants definition_entry * Evd.evar_map * 
       Universes.universe_binders * Impargs.manual_implicits
 
-val declare_definition : Id.t -> definition_kind ->
+val declare_definition : flags:Declarations.typing_flags -> Id.t -> definition_kind ->
   Safe_typing.private_constants definition_entry -> Universes.universe_binders -> Impargs.manual_implicits ->
     Globnames.global_reference Lemmas.declaration_hook -> Globnames.global_reference
 
@@ -91,6 +91,7 @@ type one_inductive_impls =
   Impargs.manual_implicits list (** for constrs *)
 
 val interp_mutual_inductive :
+  bool -> (* if [false], then positivity is assumed *)
   structured_inductive_expr -> decl_notation list -> polymorphic ->
     private_flag -> Decl_kinds.recursivity_kind ->
     mutual_inductive_entry * Universes.universe_binders * one_inductive_impls list
@@ -105,6 +106,7 @@ val declare_mutual_inductive_with_eliminations :
 (** Entry points for the vernacular commands Inductive and CoInductive *)
 
 val do_mutual_inductive :
+  bool -> (* if [false], then positivity is assumed *)
   (one_inductive_expr * decl_notation list) list -> polymorphic -> 
   private_flag -> Decl_kinds.recursivity_kind -> unit
 
@@ -148,12 +150,13 @@ val interp_cofixpoint :
 (** Registering fixpoints and cofixpoints in the environment *)
 
 val declare_fixpoint :
+  flags:Declarations.typing_flags ->
   locality -> polymorphic ->
   recursive_preentry * lident list option * Evd.evar_universe_context * 
   (Name.t list * Impargs.manual_implicits * int option) list ->
   lemma_possible_guards -> decl_notation list -> unit
 
-val declare_cofixpoint : locality -> polymorphic -> 
+val declare_cofixpoint : flags:Declarations.typing_flags -> locality -> polymorphic -> 
   recursive_preentry * lident list option * Evd.evar_universe_context * 
   (Name.t list * Impargs.manual_implicits * int option) list ->
   decl_notation list -> unit
@@ -161,14 +164,16 @@ val declare_cofixpoint : locality -> polymorphic ->
 (** Entry points for the vernacular commands Fixpoint and CoFixpoint *)
 
 val do_fixpoint :
+  flags:Declarations.typing_flags -> (* When [false], assume guarded. *)
   locality -> polymorphic -> (fixpoint_expr * decl_notation list) list -> unit
 
 val do_cofixpoint :
+  flags:Declarations.typing_flags -> (* When [false], assume guarded. *)
   locality -> polymorphic -> (cofixpoint_expr * decl_notation list) list -> unit
 
 (** Utils *)
 
 val check_mutuality : Environ.env -> bool -> (Id.t * types) list -> unit
 
-val declare_fix : ?opaque:bool -> definition_kind -> Universes.universe_binders -> Univ.universe_context -> Id.t ->
+val declare_fix : flags:Declarations.typing_flags -> ?opaque:bool -> definition_kind -> Universes.universe_binders -> Univ.universe_context -> Id.t ->
   Safe_typing.private_constants Entries.proof_output -> types -> Impargs.manual_implicits -> global_reference
diff --git a/toplevel/discharge.ml b/toplevel/discharge.ml
index 6b267283a..74361eb1c 100644
--- a/toplevel/discharge.ml
+++ b/toplevel/discharge.ml
@@ -116,5 +116,6 @@ let process_inductive (sechyps,abs_ctx) modlist mib =
     mind_entry_inds = inds';
     mind_entry_polymorphic = mib.mind_polymorphic;
     mind_entry_private = mib.mind_private;
-    mind_entry_universes = univs
+    mind_entry_universes = univs;
+    mind_entry_check_positivity = mib.mind_checked_positive;
   }
diff --git a/toplevel/indschemes.ml b/toplevel/indschemes.ml
index f995a390c..a43758e3c 100644
--- a/toplevel/indschemes.ml
+++ b/toplevel/indschemes.ml
@@ -497,7 +497,7 @@ let map_inductive_block f kn n = for i=0 to n-1 do f (kn,i) done
 let declare_default_schemes kn =
   let mib = Global.lookup_mind kn in
   let n = Array.length mib.mind_packets in
-  if !elim_flag && (mib.mind_finite <> BiFinite || !bifinite_elim_flag) then
+  if !elim_flag && (mib.mind_finite <> BiFinite || !bifinite_elim_flag) && mib.mind_checked_positive then
     declare_induction_schemes kn;
   if !case_flag then map_inductive_block declare_one_case_analysis_scheme kn n;
   if is_eq_flag() then try_declare_beq_scheme kn;
diff --git a/toplevel/obligations.ml b/toplevel/obligations.ml
index dbc3ccaac..00fd97210 100644
--- a/toplevel/obligations.ml
+++ b/toplevel/obligations.ml
@@ -567,7 +567,8 @@ let declare_mutual_definition l =
 	    List.map3 compute_possible_guardness_evidences
               wfl fixdefs fixtypes in
 	  let indexes = 
-            Pretyping.search_guard Loc.ghost (Global.env())
+              Pretyping.search_guard ~tflags:{Declarations.check_guarded=true}
+              Loc.ghost (Global.env())
               possible_indexes fixdecls in
           Some indexes, 
           List.map_i (fun i _ ->
diff --git a/toplevel/record.ml b/toplevel/record.ml
index 0c3bd953c..214d44d83 100644
--- a/toplevel/record.ml
+++ b/toplevel/record.ml
@@ -361,7 +361,7 @@ let structure_signature ctx =
 
 open Typeclasses
 
-let declare_structure finite poly ctx id idbuild paramimpls params arity template 
+let declare_structure chk finite poly ctx id idbuild paramimpls params arity template 
     fieldimpls fields ?(kind=StructureComponent) ?name is_coe coers sign =
   let nparams = List.length params and nfields = List.length fields in
   let args = Context.Rel.to_extended_list nfields params in
@@ -386,7 +386,8 @@ let declare_structure finite poly ctx id idbuild paramimpls params arity templat
       mind_entry_inds = [mie_ind];
       mind_entry_polymorphic = poly;
       mind_entry_private = None;
-      mind_entry_universes = ctx } in
+      mind_entry_universes = ctx;
+      mind_entry_check_positivity = chk; } in
   let kn = Command.declare_mutual_inductive_with_eliminations mie [] [(paramimpls,[])] in
   let rsp = (kn,0) in (* This is ind path of idstruc *)
   let cstr = (rsp,1) in
@@ -405,7 +406,7 @@ let implicits_of_context ctx =
     in ExplByPos (i, explname), (true, true, true))
     1 (List.rev (Anonymous :: (List.map get_name ctx)))
 
-let declare_class finite def poly ctx id idbuild paramimpls params arity 
+let declare_class chk finite def poly ctx id idbuild paramimpls params arity 
     template fieldimpls fields ?(kind=StructureComponent) is_coe coers priorities sign =
   let fieldimpls =
     (* Make the class implicit in the projections, and the params if applicable. *)
@@ -448,7 +449,7 @@ let declare_class finite def poly ctx id idbuild paramimpls params arity
       in
       cref, [Name proj_name, sub, Some proj_cst]
     | _ ->
-       let ind = declare_structure BiFinite poly ctx (snd id) idbuild paramimpls
+       let ind = declare_structure chk BiFinite poly ctx (snd id) idbuild paramimpls
 	  params arity template fieldimpls fields
 	  ~kind:Method ~name:binder_name false (List.map (fun _ -> false) fields) sign
        in
@@ -527,8 +528,9 @@ open Vernacexpr
 
 (* [fs] corresponds to fields and [ps] to parameters; [coers] is a
    list telling if the corresponding fields must me declared as coercions 
-   or subinstances *)
-let definition_structure (kind,poly,finite,(is_coe,((loc,idstruc),pl)),ps,cfs,idbuild,s) =
+   or subinstances. When [chk] is false positivity is
+    assumed. *)
+let definition_structure chk (kind,poly,finite,(is_coe,((loc,idstruc),pl)),ps,cfs,idbuild,s) =
   let cfs,notations = List.split cfs in
   let cfs,priorities = List.split cfs in
   let coers,fs = List.split cfs in
@@ -549,14 +551,14 @@ let definition_structure (kind,poly,finite,(is_coe,((loc,idstruc),pl)),ps,cfs,id
   let sign = structure_signature (fields@params) in
   let gr = match kind with
     | Class def ->
-	let gr = declare_class finite def poly ctx (loc,idstruc) idbuild
+	let gr = declare_class chk finite def poly ctx (loc,idstruc) idbuild
 	  implpars params arity template implfs fields is_coe coers priorities sign in
 	gr
     | _ ->
 	let implfs = List.map
 	  (fun impls -> implpars @ Impargs.lift_implicits
 	    (succ (List.length params)) impls) implfs in
-	let ind = declare_structure finite poly ctx idstruc 
+	let ind = declare_structure chk finite poly ctx idstruc 
 	  idbuild implpars params arity template implfs 
 	  fields is_coe (List.map (fun coe -> not (Option.is_empty coe)) coers) sign in
 	IndRef ind
diff --git a/toplevel/record.mli b/toplevel/record.mli
index 26eb3378b..525326237 100644
--- a/toplevel/record.mli
+++ b/toplevel/record.mli
@@ -24,7 +24,9 @@ val declare_projections :
   coercion_flag list -> manual_explicitation list list -> Context.Rel.t ->
   (Name.t * bool) list * constant option list
 
-val declare_structure : Decl_kinds.recursivity_kind ->
+val declare_structure :
+  bool -> (** check positivity? *)
+  Decl_kinds.recursivity_kind ->
   bool (** polymorphic?*) -> Univ.universe_context ->
   Id.t -> Id.t ->
   manual_explicitation list -> Context.Rel.t -> (** params *) constr -> (** arity *)
@@ -37,6 +39,7 @@ val declare_structure : Decl_kinds.recursivity_kind ->
   inductive
 
 val definition_structure :
+  bool -> (** check positivity? *)
   inductive_kind * Decl_kinds.polymorphic * Decl_kinds.recursivity_kind * plident with_coercion * local_binder list *
   (local_decl_expr with_instance with_priority with_notation) list *
   Id.t * constr_expr option -> global_reference
diff --git a/toplevel/vernacentries.ml b/toplevel/vernacentries.ml
index 222b7d3df..0b4dc0d18 100644
--- a/toplevel/vernacentries.ml
+++ b/toplevel/vernacentries.ml
@@ -519,7 +519,7 @@ let vernac_assumption locality poly (local, kind) l nl =
   let status = do_assumptions kind nl l in
   if not status then Feedback.feedback Feedback.AddedAxiom
 
-let vernac_record k poly finite struc binders sort nameopt cfs =
+let vernac_record chk k poly finite struc binders sort nameopt cfs =
   let const = match nameopt with
     | None -> add_prefix "Build_" (snd (fst (snd struc)))
     | Some (_,id as lid) ->
@@ -530,9 +530,14 @@ let vernac_record k poly finite struc binders sort nameopt cfs =
 	match x with
 	| Vernacexpr.AssumExpr ((loc, Name id), _) -> Dumpglob.dump_definition (loc,id) false "proj"
 	| _ -> ()) cfs);
-    ignore(Record.definition_structure (k,poly,finite,struc,binders,cfs,const,sort))
-
-let vernac_inductive poly lo finite indl =
+    ignore(Record.definition_structure chk (k,poly,finite,struc,binders,cfs,const,sort))
+
+(** When [chk] is false, positivity is assumed. When [poly] is true
+    the type is declared polymorphic. When [lo] is true, then the type
+    is declared private (as per the [Private] keyword). [finite]
+    indicates whether the type is inductive, co-inductive or
+    neither. *)
+let vernac_inductive chk poly lo finite indl =
   if Dumpglob.dump () then
     List.iter (fun (((coe,(lid,_)), _, _, _, cstrs), _) ->
       match cstrs with
@@ -548,14 +553,14 @@ let vernac_inductive poly lo finite indl =
   | [ (_ , _ , _ ,Variant, RecordDecl _),_ ] ->
       Errors.error "The Variant keyword cannot be used to define a record type. Use Record instead."
   | [ ( id , bl , c , b, RecordDecl (oc,fs) ), [] ] ->
-      vernac_record (match b with Class true -> Class false | _ -> b)
+      vernac_record chk (match b with Class true -> Class false | _ -> b)
        poly finite id bl c oc fs
   | [ ( id , bl , c , Class true, Constructors [l]), _ ] ->
       let f =
 	let (coe, ((loc, id), ce)) = l in
 	let coe' = if coe then Some true else None in
   	  (((coe', AssumExpr ((loc, Name id), ce)), None), [])
-      in vernac_record (Class true) poly finite id bl c None [f]
+      in vernac_record chk (Class true) poly finite id bl c None [f]
   | [ ( id , bl , c , Class true, _), _ ] ->
       Errors.error "Definitional classes must have a single method"
   | [ ( id , bl , c , Class false, Constructors _), _ ] ->
@@ -569,19 +574,19 @@ let vernac_inductive poly lo finite indl =
       | _ -> Errors.error "Cannot handle mutually (co)inductive records."
     in
     let indl = List.map unpack indl in
-    do_mutual_inductive indl poly lo finite
+    do_mutual_inductive chk indl poly lo finite
 
-let vernac_fixpoint locality poly local l =
+let vernac_fixpoint ~flags locality poly local l =
   let local = enforce_locality_exp locality local in
   if Dumpglob.dump () then
     List.iter (fun (((lid,_), _, _, _, _), _) -> Dumpglob.dump_definition lid false "def") l;
-  do_fixpoint local poly l
+  do_fixpoint ~flags local poly l
 
-let vernac_cofixpoint locality poly local l =
+let vernac_cofixpoint ~flags locality poly local l =
   let local = enforce_locality_exp locality local in
   if Dumpglob.dump () then
     List.iter (fun (((lid,_), _, _, _), _) -> Dumpglob.dump_definition lid false "def") l;
-  do_cofixpoint local poly l
+  do_cofixpoint ~flags local poly l
 
 let vernac_scheme l =
   if Dumpglob.dump () then
@@ -1726,6 +1731,7 @@ let vernac_load interp fname =
   try while true do interp (snd (parse_sentence input)) done
   with End_of_input -> ()
 
+let all_checks = { Declarations.check_guarded = true }
 
 (* "locality" is the prefix "Local" attribute, while the "local" component
  * is the outdated/deprecated "Local" attribute of some vernacular commands
@@ -1763,9 +1769,9 @@ let interp ?proof ~loc locality poly c =
   | VernacEndProof e -> vernac_end_proof ?proof e
   | VernacExactProof c -> vernac_exact_proof c
   | VernacAssumption (stre,nl,l) -> vernac_assumption locality poly stre l nl
-  | VernacInductive (priv,finite,l) -> vernac_inductive poly priv finite l
-  | VernacFixpoint (local, l) -> vernac_fixpoint locality poly local l
-  | VernacCoFixpoint (local, l) -> vernac_cofixpoint locality poly local l
+  | VernacInductive (priv,finite,l) -> vernac_inductive true poly priv finite l
+  | VernacFixpoint (local, l) -> vernac_fixpoint locality ~flags:all_checks poly local l
+  | VernacCoFixpoint (local, l) -> vernac_cofixpoint ~flags:all_checks locality poly local l
   | VernacScheme l -> vernac_scheme l
   | VernacCombinedScheme (id, l) -> vernac_combined_scheme id l
   | VernacUniverse l -> vernac_universe loc poly l