diff options
Diffstat (limited to 'kernel/indtypes.ml')
| -rw-r--r-- | kernel/indtypes.ml | 78 |
1 files changed, 60 insertions, 18 deletions
diff --git a/kernel/indtypes.ml b/kernel/indtypes.ml index a46c33bf0..21d1e7134 100644 --- a/kernel/indtypes.ml +++ b/kernel/indtypes.ml @@ -30,8 +30,13 @@ let indices_matter = ref false let enforce_indices_matter () = indices_matter := true let is_indices_matter () = !indices_matter -(* Same as noccur_between but may perform reductions. - Could be refined more... *) +(* [weaker_noccur_between env n nvars t] (defined above), checks that + no de Bruijn indices between [n] and [n+nvars] occur in [t]. If + some such occurrences are found, then reduction is performed + (lazily for efficiency purposes) in order to determine whether + these occurrences are occurrences in the normal form. If the + occurrences are eliminated a witness reduct [Some t'] of [t] is + returned otherwise [None] is returned. *) let weaker_noccur_between env x nvars t = if noccur_between x nvars t then Some t else @@ -284,7 +289,7 @@ let typecheck_inductive env mie = let full_polymorphic () = let defu = Term.univ_of_sort def_level in let is_natural = - type_in_type env || (check_leq (universes env') infu defu) + type_in_type env || (UGraph.check_leq (universes env') infu defu) in let _ = (** Impredicative sort, always allow *) @@ -310,7 +315,7 @@ let typecheck_inductive env mie = (* conclusions of the parameters *) (* We enforce [u >= lev] in case [lev] has a strict upper *) (* constraints over [u] *) - let b = type_in_type env || check_leq (universes env') infu u in + let b = type_in_type env || UGraph.check_leq (universes env') infu u in if not b then anomaly ~label:"check_inductive" (Pp.str"Incorrect universe " ++ @@ -451,17 +456,30 @@ let rec ienv_decompose_prod (env,_,_,_ as ienv) n c = let array_min nmr a = if Int.equal nmr 0 then 0 else Array.fold_left (fun k (nmri,_) -> min k nmri) nmr a -(* The recursive function that checks positivity and builds the list - of recursive arguments *) +(** [check_positivity_one ienv hyps (mind,i) nargs lcnames indlc] + checks the positivity of the [i]-th member of the mutually + inductive definition [mind]. It returns an [Rtree.t] which + represents the position of the recursive calls of inductive in [i] + 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 (env,_,ntypes,_ as ienv) hyps (_,i as ind) nargs lcnames indlc = let lparams = rel_context_length hyps in let nmr = rel_context_nhyps hyps in - (* Checking the (strict) positivity of a constructor argument type [c] *) + (** Positivity of one argument [c] of a constructor (i.e. the + constructor [cn] has a type of the shape [… -> c … -> P], where, + more generally, the arrows may be dependent). *) let rec check_pos (env, n, ntypes, ra_env as ienv) nmr c = let x,largs = decompose_app (whd_betadeltaiota env c) in match kind_of_term x with | Prod (na,b,d) -> let () = assert (List.is_empty largs) in + (** If one of the inductives of the mutually inductive + block occurs in the left-hand side of a product, then + such an occurrence is a non-strictly-positive + 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] | Some b -> @@ -474,21 +492,35 @@ let check_positivity_one (env,_,ntypes,_ as ienv) hyps (_,i as ind) nargs lcname Mrec _ -> compute_rec_par ienv hyps nmr largs | _ -> nmr) in + (** 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) then failwith_non_pos_list n ntypes largs else (nmr1,rarg) with Failure _ | Invalid_argument _ -> (nmr,mk_norec)) | Ind ind_kn -> - (* If the inductive type being defined appears in a - parameter, then we have a nested indtype *) + (** If one of the inductives of the mutually inductive + block being defined appears in a parameter, then we + have a nested inductive type. The positivity is then + discharged to the [check_positive_nested] function. *) if List.for_all (noccur_between n ntypes) largs then (nmr,mk_norec) else check_positive_nested ienv nmr (ind_kn, largs) | err -> + (** 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 then (nmr,mk_norec) else failwith_non_pos_list n ntypes (x::largs) + (** [check_positive_nested] handles the case of nested inductive + calls, that is, when an inductive types from the mutually + inductive block is called as an argument of an inductive types + (for the moment, this inductive type must be a previously + defined types, not one of the types of the mutually inductive + block being defined). *) (* accesses to the environment are not factorised, but is it worth? *) and check_positive_nested (env,n,ntypes,ra_env as ienv) nmr ((mi,u), largs) = let (mib,mip) = lookup_mind_specif env mi in @@ -497,12 +529,13 @@ let check_positivity_one (env,_,ntypes,_ as ienv) hyps (_,i as ind) nargs lcname let (lpar,auxlargs) = try List.chop auxnpar largs with Failure _ -> raise (IllFormedInd (LocalNonPos n)) in - (* If the inductive appears in the args (non params) then the - definition is not positive. *) + (** 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) auxlargs) then failwith_non_pos_list n ntypes auxlargs; - (* We do not deal with imbricated mutual inductive types *) + (* Nested mutual inductive types are not supported *) let auxntyp = mib.mind_ntypes in if not (Int.equal auxntyp 1) then raise (IllFormedInd (LocalNonPos n)); (* The nested inductive type with parameters removed *) @@ -513,8 +546,11 @@ let check_positivity_one (env,_,ntypes,_ as ienv) hyps (_,i as ind) nargs lcname (* Parameters expressed in env' *) let lpar' = List.map (lift auxntyp) lpar in let irecargs_nmr = - (* fails if the inductive type occurs non positively *) - (* with recursive parameters substituted *) + (** Checks that the "nesting" inductive type is covariant in + the relevant parameters. In other words, that the + (nested) parameters which are instantiated with + inductives of the mutually inductive block occur + positively in the types of the nested constructors. *) Array.map (function c -> let c' = hnf_prod_applist env' c lpar' in @@ -528,10 +564,14 @@ let check_positivity_one (env,_,ntypes,_ as ienv) hyps (_,i as ind) nargs lcname in (nmr',(Rtree.mk_rec [|mk_paths (Imbr mi) irecargs|]).(0)) - (* check the inductive types occur positively in the products of C, if - check_head=true, also check the head corresponds to a constructor of - the ith type *) - + (** [check_constructors ienv check_head nmr c] checks the positivity + condition in the type [c] of a constructor (i.e. that recursive + calls to the inductives of the mutually inductive definition + appear strictly positively in each of the arguments of the + constructor, see also [check_pos]). If [check_head] is [true], + then the type of the fully applied constructor (the "head" of + the type [c]) is checked to be the right (properly applied) + inductive type. *) and check_constructors ienv check_head nmr c = let rec check_constr_rec (env,n,ntypes,ra_env as ienv) nmr lrec c = let x,largs = decompose_app (whd_betadeltaiota env c) in @@ -571,6 +611,8 @@ let check_positivity_one (env,_,ntypes,_ as ienv) hyps (_,i as ind) nargs lcname and nmr' = array_min nmr irecargs_nmr in (nmr', mk_paths (Mrec ind) irecargs) +(** [check_positivity kn env_ar params] checks that the mutually + inductive block [inds] is strictly positive. *) let check_positivity kn env_ar params inds = let ntypes = Array.length inds in let rc = Array.mapi (fun j t -> (Mrec (kn,j),t)) (Rtree.mk_rec_calls ntypes) in |