(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* = Array.length mib.mind_packets then error "Inductive.lookup_mind_specif: invalid inductive index"; (mib, mib.mind_packets.(tyi)) let find_rectype env c = let (t, l) = decompose_app (whd_betadeltaiota env c) in match kind_of_term t with | Ind ind -> (ind, l) | _ -> raise Not_found let find_inductive env c = let (t, l) = decompose_app (whd_betadeltaiota env c) in match kind_of_term t with | Ind ind when (fst (lookup_mind_specif env ind)).mind_finite -> (ind, l) | _ -> raise Not_found let find_coinductive env c = let (t, l) = decompose_app (whd_betadeltaiota env c) in match kind_of_term t with | Ind ind when not (fst (lookup_mind_specif env ind)).mind_finite -> (ind, l) | _ -> raise Not_found (************************************************************************) (* Build the substitution that replaces Rels by the appropriate *) (* inductives *) let ind_subst mind mib = let ntypes = mib.mind_ntypes in let make_Ik k = mkInd (mind,ntypes-k-1) in list_tabulate make_Ik ntypes (* Instantiate inductives in constructor type *) let constructor_instantiate mind mib c = let s = ind_subst mind mib in substl s c let instantiate_params full t args sign = let fail () = anomaly "instantiate_params: type, ctxt and args mismatch" in let (rem_args, subs, ty) = Sign.fold_rel_context (fun (_,copt,_) (largs,subs,ty) -> match (copt, largs, kind_of_term ty) with | (None, a::args, Prod(_,_,t)) -> (args, a::subs, t) | (Some b,_,LetIn(_,_,_,t)) -> (largs, (substl subs b)::subs, t) | (_,[],_) -> if full then fail() else ([], subs, ty) | _ -> fail ()) sign ~init:(args,[],t) in if rem_args <> [] then fail(); substl subs ty let instantiate_partial_params = instantiate_params false let full_inductive_instantiate mib params t = instantiate_params true t params mib.mind_params_ctxt let full_constructor_instantiate (((mind,_),mib,_),params) = let inst_ind = constructor_instantiate mind mib in (fun t -> instantiate_params true (inst_ind t) params mib.mind_params_ctxt) (************************************************************************) (************************************************************************) (* Functions to build standard types related to inductive *) (* For each inductive type of a block that is of level u_i, we have the constraints that u_i >= v_i where v_i is the type level of the types of the constructors of this inductive type. Each v_i depends of some of the u_i and of an extra (maybe non variable) universe, say w_i. Typically, for three inductive types, we could have u1,u2,u3,w1 <= u1 u1 w2 <= u2 u2,u3,w3 <= u3 From this system of inequations, we shall deduce w1,w2,w3 <= u1 w1,w2 <= u2 w1,w2,w3 <= u3 *) let number_of_inductives mib = Array.length mib.mind_packets let number_of_constructors mip = Array.length mip.mind_consnames (* Computing the actual sort of an applied or partially applied inductive type: I_i: forall uniformparams:utyps, forall otherparams:otyps, Type(a) uniformargs : utyps otherargs : otyps I_1:forall ...,s_1;...I_n:forall ...,s_n |- sort(C_kj(uniformargs)) = s_kj s'_k = max(..s_kj..) merge(..s'_k..) = ..s''_k.. -------------------------------------------------------------------- Gamma |- I_i uniformargs otherargs : phi(s''_i) where - if p=0, phi() = Prop - if p=1, phi(s) = s - if p<>1, phi(s) = sup(Set,s) Remark: Set (predicative) is encoded as Type(0) *) let find_constraint levels level_bounds i nci = if nci = 0 then mk_Prop else let level_bounds' = solve_constraints_system levels level_bounds in let level = level_bounds'.(i) in if nci = 1 & is_empty_universe level then mk_Prop else if Univ.is_base level then mk_Set else Type level let find_inductive_level env (mib,mip) (_,i) levels level_bounds = find_constraint levels level_bounds i (number_of_constructors mip) let set_inductive_level env s t = let sign,s' = dest_prod_assum env t in if family_of_sort s <> family_of_sort (destSort s') then (* This induces reductions if user_arity <> nf_arity *) mkArity (sign,s) else t let constructor_instances env (mib,mip) (_,i) args = let nargs = Array.length args in let args = Array.to_list args in let uargs = if nargs > mib.mind_nparams_rec then fst (list_chop mib.mind_nparams_rec args) else args in let arities = Array.map (fun mip -> destArity mip.mind_nf_arity) mib.mind_packets in (* Compute the minimal type *) let levels = Array.init (number_of_inductives mib) (fun _ -> fresh_local_univ ()) in let arities = list_tabulate (fun i -> let ctx,s = arities.(i) in let s = match s with Type _ -> Type (levels.(i)) | s -> s in (Name mib.mind_packets.(i).mind_typename,None,mkArity (ctx,s))) (number_of_inductives mib) in (* Remark: all arities are closed hence no need for lift *) let env_ar = push_rel_context (List.rev arities) env in let uargs = List.map (lift (number_of_inductives mib)) uargs in let lc = Array.map (fun mip -> Array.map (fun c -> instantiate_partial_params c uargs mib.mind_params_ctxt) mip.mind_nf_lc) mib.mind_packets in env_ar, lc, levels let is_small_inductive (mip,mib) = is_small (snd (destArity mib.mind_nf_arity)) let max_inductive_sort v = let v = Array.map (function | Type u -> u | _ -> anomaly "Only type levels when computing the minimal sort of an inductive type") v in Univ.sup_array v (* Type of an inductive type *) let type_of_inductive (_,mip) = mip.mind_user_arity (************************************************************************) (* Type of a constructor *) let type_of_constructor cstr (mib,mip) = let ind = inductive_of_constructor cstr in let specif = mip.mind_user_lc in let i = index_of_constructor cstr in let nconstr = Array.length mip.mind_consnames in if i > nconstr then error "Not enough constructors in the type"; constructor_instantiate (fst ind) mib specif.(i-1) let arities_of_specif kn (mib,mip) = let specif = mip.mind_nf_lc in Array.map (constructor_instantiate kn mib) specif let arities_of_constructors ind specif = arities_of_specif (fst ind) specif (************************************************************************) let is_info_arity env c = match dest_arity env c with | (_,Prop Null) -> false | (_,Prop Pos) -> true | (_,Type _) -> true let error_elim_expln env kp ki = if is_info_arity env kp && not (is_info_arity env ki) then NonInformativeToInformative else match (kind_of_term kp,kind_of_term ki) with | Sort (Type _), Sort (Prop _) -> StrongEliminationOnNonSmallType | _ -> WrongArity (* Type of case predicates *) let local_rels ctxt = let (rels,_) = Sign.fold_rel_context_reverse (fun (rels,n) (_,copt,_) -> match copt with None -> (mkRel n :: rels, n+1) | Some _ -> (rels, n+1)) ~init:([],1) ctxt in rels (* Get type of inductive, with parameters instantiated *) let get_arity mib mip params = let arity = mip.mind_nf_arity in destArity (full_inductive_instantiate mib params arity) let rel_list n m = let rec reln l p = if p>m then l else reln (mkRel(n+p)::l) (p+1) in reln [] 1 let build_dependent_inductive ind mib mip params = let nrealargs = mip.mind_nrealargs in applist (mkInd ind, (List.map (lift nrealargs) params)@(rel_list 0 nrealargs)) (* This exception is local *) exception LocalArity of (constr * constr * arity_error) option let is_correct_arity env c pj ind mib mip params = let kelim = mip.mind_kelim in let arsign,s = get_arity mib mip params in let nodep_ar = it_mkProd_or_LetIn (mkSort s) arsign in let rec srec env pt t u = let pt' = whd_betadeltaiota env pt in let t' = whd_betadeltaiota env t in match kind_of_term pt', kind_of_term t' with | Prod (na1,a1,a2), Prod (_,a1',a2') -> let univ = try conv env a1 a1' with NotConvertible -> raise (LocalArity None) in srec (push_rel (na1,None,a1) env) a2 a2' (Constraint.union u univ) | Prod (_,a1,a2), _ -> let k = whd_betadeltaiota env a2 in let ksort = match kind_of_term k with | Sort s -> family_of_sort s | _ -> raise (LocalArity None) in let dep_ind = build_dependent_inductive ind mib mip params in let univ = try conv env a1 dep_ind with NotConvertible -> raise (LocalArity None) in if List.exists ((=) ksort) kelim then (true, Constraint.union u univ) else raise (LocalArity (Some(k,t',error_elim_expln env k t'))) | k, Prod (_,_,_) -> raise (LocalArity None) | k, ki -> let ksort = match k with | Sort s -> family_of_sort s | _ -> raise (LocalArity None) in if List.exists ((=) ksort) kelim then (false, u) else raise (LocalArity (Some(pt',t',error_elim_expln env pt' t'))) in try srec env pj.uj_type nodep_ar Constraint.empty with LocalArity kinds -> let create_sort = function | InProp -> mkProp | InSet -> mkSet | InType -> mkSort type_0 in let listarity = List.map create_sort kelim (* let listarity = (List.map (fun s -> make_arity env true indf (create_sort s)) kelim) @(List.map (fun s -> make_arity env false indf (create_sort s)) kelim)*) in error_elim_arity env ind listarity c pj kinds (************************************************************************) (* Type of case branches *) (* [p] is the predicate, [i] is the constructor number (starting from 0), and [cty] is the type of the constructor (params not instantiated) *) let build_branches_type ind mib mip params dep p = let build_one_branch i cty = let typi = full_constructor_instantiate ((ind,mib,mip),params) cty in let (args,ccl) = decompose_prod_assum typi in let nargs = rel_context_length args in let (_,allargs) = decompose_app ccl in let (lparams,vargs) = list_chop mib.mind_nparams allargs in let cargs = if dep then let cstr = ith_constructor_of_inductive ind (i+1) in let dep_cstr = applist (mkConstruct cstr,lparams@(local_rels args)) in vargs @ [dep_cstr] else vargs in let base = beta_appvect (lift nargs p) (Array.of_list cargs) in it_mkProd_or_LetIn base args in Array.mapi build_one_branch mip.mind_nf_lc (* [p] is the predicate, [c] is the match object, [realargs] is the list of real args of the inductive type *) let build_case_type dep p c realargs = let args = if dep then realargs@[c] else realargs in beta_appvect p (Array.of_list args) let type_case_branches env (ind,largs) pj c = let (mib,mip) = lookup_mind_specif env ind in let nparams = mib.mind_nparams in let (params,realargs) = list_chop nparams largs in let p = pj.uj_val in let (dep,univ) = is_correct_arity env c pj ind mib mip params in let lc = build_branches_type ind mib mip params dep p in let ty = build_case_type dep p c realargs in (lc, ty, univ) (************************************************************************) (* Checking the case annotation is relevent *) let rec inductive_kn_equiv env kn1 kn2 = match (lookup_mind kn1 env).mind_equiv with | Some kn1' -> inductive_kn_equiv env kn2 kn1' | None -> match (lookup_mind kn2 env).mind_equiv with | Some kn2' -> inductive_kn_equiv env kn2' kn1 | None -> false let inductive_equiv env (kn1,i1) (kn2,i2) = i1=i2 & inductive_kn_equiv env kn1 kn2 let check_case_info env indsp ci = let (mib,mip) = lookup_mind_specif env indsp in if (indsp <> ci.ci_ind) or (mib.mind_nparams <> ci.ci_npar) or (mip.mind_consnrealdecls <> ci.ci_cstr_nargs) then raise (TypeError(env,WrongCaseInfo(indsp,ci))) (************************************************************************) (************************************************************************) (* Guard conditions for fix and cofix-points *) (* Check if t is a subterm of Rel n, and gives its specification, assuming lst already gives index of subterms with corresponding specifications of recursive arguments *) (* A powerful notion of subterm *) (* To each inductive definition corresponds an array describing the structure of recursive arguments for each constructor, we call it the recursive spec of the type (it has type recargs vect). For checking the guard, we start from the decreasing argument (Rel n) with its recursive spec. During checking the guardness condition, we collect patterns variables corresponding to subterms of n, each of them with its recursive spec. They are organised in a list lst of type (int * recargs) list which is sorted with respect to the first argument. *) (*************************) (* Environment annotated with marks on recursive arguments: it is a triple (env,lst,n) where - env is the typing environment - lst is a mapping from de Bruijn indices to list of recargs (tells which subterms of that variable are recursive) - n is the de Bruijn index of the fixpoint for which we are checking the guard condition. Below are functions to handle such environment. *) type size = Large | Strict let size_glb s1 s2 = match s1,s2 with Strict, Strict -> Strict | _ -> Large type subterm_spec = Subterm of (size * wf_paths) | Dead_code | Not_subterm let spec_of_tree t = if t=mk_norec then Not_subterm else Subterm(Strict,t) let subterm_spec_glb = let glb2 s1 s2 = match s1,s2 with _, Dead_code -> s1 | Dead_code, _ -> s2 | Not_subterm, _ -> Not_subterm | _, Not_subterm -> Not_subterm | Subterm (a1,t1), Subterm (a2,t2) -> if t1=t2 then Subterm (size_glb a1 a2, t1) (* branches do not return objects with same spec *) else Not_subterm in Array.fold_left glb2 Dead_code type guard_env = { env : env; (* dB of last fixpoint *) rel_min : int; (* inductive of recarg of each fixpoint *) inds : inductive array; (* the recarg information of inductive family *) recvec : wf_paths array; (* dB of variables denoting subterms *) genv : subterm_spec list; } let make_renv env minds recarg (kn,tyi) = let mib = Environ.lookup_mind kn env in let mind_recvec = Array.map (fun mip -> mip.mind_recargs) mib.mind_packets in { env = env; rel_min = recarg+2; inds = minds; recvec = mind_recvec; genv = [Subterm(Large,mind_recvec.(tyi))] } let push_var renv (x,ty,spec) = { renv with env = push_rel (x,None,ty) renv.env; rel_min = renv.rel_min+1; genv = spec:: renv.genv } let assign_var_spec renv (i,spec) = { renv with genv = list_assign renv.genv (i-1) spec } let push_var_renv renv (x,ty) = push_var renv (x,ty,Not_subterm) (* Fetch recursive information about a variable p *) let subterm_var p renv = try List.nth renv.genv (p-1) with Failure _ | Invalid_argument _ -> Not_subterm (* Add a variable and mark it as strictly smaller with information [spec]. *) let add_subterm renv (x,a,spec) = push_var renv (x,a,spec_of_tree spec) let push_ctxt_renv renv ctxt = let n = rel_context_length ctxt in { renv with env = push_rel_context ctxt renv.env; rel_min = renv.rel_min+n; genv = iterate (fun ge -> Not_subterm::ge) n renv.genv } let push_fix_renv renv (_,v,_ as recdef) = let n = Array.length v in { renv with env = push_rec_types recdef renv.env; rel_min = renv.rel_min+n; genv = iterate (fun ge -> Not_subterm::ge) n renv.genv } (******************************) (* Computing the recursive subterms of a term (propagation of size information through Cases). *) (* c is a branch of an inductive definition corresponding to the spec lrec. mind_recvec is the recursive spec of the inductive definition of the decreasing argument n. case_branches_specif renv lrec lc will pass the lambdas of c corresponding to pattern variables and collect possibly new subterms variables and returns the bodies of the branches with the correct envs and decreasing args. *) let lookup_subterms env ind = let (_,mip) = lookup_mind_specif env ind in mip.mind_recargs (*********************************) (* finds the inductive type of the recursive argument of a fixpoint *) let inductive_of_fix env recarg body = let (ctxt,b) = decompose_lam_n_assum recarg body in let env' = push_rel_context ctxt env in let (_,ty,_) = destLambda(whd_betadeltaiota env' b) in let (i,_) = decompose_app (whd_betadeltaiota env' ty) in destInd i (* subterm_specif env c ind subterm_specif should test if [c] (building objects of inductive type [ind], not necessarily the same as that of the recursive argument) is a subterm of the recursive argument of the fixpoint we are checking and fails with Not_found if not. In case it is, it should send its recursive specification (i.e. on which arguments we are allowed to make recursive calls). This recursive spec should be the same size as the number of constructors of the type of c. Returns: - [Some lc] if [c] is a strict subterm of the rec. arg. (or a Meta) - [None] otherwise *) let rec subterm_specif renv t ind = let f,l = decompose_app (whd_betadeltaiota renv.env t) in match kind_of_term f with | Rel k -> subterm_var k renv | Case (ci,_,c,lbr) -> if Array.length lbr = 0 then Dead_code else let lbr_spec = case_branches_specif renv c ci.ci_ind lbr in let stl = Array.map (fun (renv',br') -> subterm_specif renv' br' ind) lbr_spec in subterm_spec_glb stl | Fix ((recindxs,i),(_,typarray,bodies as recdef)) -> (* when proving that the fixpoint f(x)=e is less than n, it is enough to prove that e is less than n assuming f is less than n furthermore when f is applied to a term which is strictly less than n, one may assume that x itself is strictly less than n *) let nbfix = Array.length typarray in let recargs = lookup_subterms renv.env ind in (* pushing the fixpoints *) let renv' = push_fix_renv renv recdef in let renv' = assign_var_spec renv' (nbfix-i, Subterm(Strict,recargs)) in let decrArg = recindxs.(i) in let theBody = bodies.(i) in let nbOfAbst = decrArg+1 in let sign,strippedBody = decompose_lam_n_assum nbOfAbst theBody in (* pushing the fix parameters *) let renv'' = push_ctxt_renv renv' sign in let renv'' = if List.length l < nbOfAbst then renv'' else let decrarg_ind = inductive_of_fix renv''.env decrArg theBody in let theDecrArg = List.nth l decrArg in let arg_spec = subterm_specif renv theDecrArg decrarg_ind in assign_var_spec renv'' (1, arg_spec) in subterm_specif renv'' strippedBody ind | Lambda (x,a,b) -> assert (l=[]); subterm_specif (push_var_renv renv (x,a)) b ind (* A term with metas is considered OK *) | Meta _ -> Dead_code (* Other terms are not subterms *) | _ -> Not_subterm (* Propagation of size information through Cases: if the matched object is a recursive subterm then compute the information associated to its own subterms. Rq: if branch is not eta-long, then the recursive information is not propagated *) and case_branches_specif renv c ind lbr = let c_spec = subterm_specif renv c ind in let rec push_branch_args renv lrec c = let c' = strip_outer_cast (whd_betadeltaiota renv.env c) in match lrec, kind_of_term c' with | (ra::lr,Lambda (x,a,b)) -> let renv' = push_var renv (x,a,ra) in push_branch_args renv' lr b | (_,_) -> (renv,c') in match c_spec with Subterm (_,t) -> let sub_spec = Array.map (List.map spec_of_tree) (dest_subterms t) in assert (Array.length sub_spec = Array.length lbr); array_map2 (push_branch_args renv) sub_spec lbr | Dead_code -> let t = dest_subterms (lookup_subterms renv.env ind) in let sub_spec = Array.map (List.map (fun _ -> Dead_code)) t in assert (Array.length sub_spec = Array.length lbr); array_map2 (push_branch_args renv) sub_spec lbr | Not_subterm -> Array.map (fun c -> (renv,c)) lbr (* Check term c can be applied to one of the mutual fixpoints. *) let check_is_subterm renv c ind = match subterm_specif renv c ind with Subterm (Strict,_) | Dead_code -> true | _ -> false (************************************************************************) exception FixGuardError of env * guard_error let error_illegal_rec_call renv fx arg = let (_,le_vars,lt_vars) = List.fold_left (fun (i,le,lt) sbt -> match sbt with (Subterm(Strict,_) | Dead_code) -> (i+1, le, i::lt) | (Subterm(Large,_)) -> (i+1, i::le, lt) | _ -> (i+1, le ,lt)) (1,[],[]) renv.genv in raise (FixGuardError (renv.env, RecursionOnIllegalTerm(fx,arg,le_vars,lt_vars))) let error_partial_apply renv fx = raise (FixGuardError (renv.env,NotEnoughArgumentsForFixCall fx)) (* Check if [def] is a guarded fixpoint body with decreasing arg. given [recpos], the decreasing arguments of each mutually defined fixpoint. *) let check_one_fix renv recpos def = let nfi = Array.length recpos in let rec check_rec_call renv t = (* if [t] does not make recursive calls, it is guarded: *) noccur_with_meta renv.rel_min nfi t or (* Rq: why not try and expand some definitions ? *) let f,l = decompose_app (whd_betaiotazeta renv.env t) in match kind_of_term f with | Rel p -> (* Test if it is a recursive call: *) if renv.rel_min <= p & p < renv.rel_min+nfi then (* the position of the invoked fixpoint: *) let glob = renv.rel_min+nfi-1-p in (* the decreasing arg of the rec call: *) let np = recpos.(glob) in if List.length l <= np then error_partial_apply renv glob; match list_chop np l with (la,(z::lrest)) -> (* Check the decreasing arg is smaller *) if not (check_is_subterm renv z renv.inds.(glob)) then error_illegal_rec_call renv glob z; List.for_all (check_rec_call renv) (la@lrest) | _ -> assert false (* otherwise check the arguments are guarded: *) else List.for_all (check_rec_call renv) l | Case (ci,p,c_0,lrest) -> List.for_all (check_rec_call renv) (c_0::p::l) && (* compute the recarg information for the arguments of each branch *) let lbr = case_branches_specif renv c_0 ci.ci_ind lrest in array_for_all (fun (renv',br') -> check_rec_call renv' br') lbr (* Enables to traverse Fixpoint definitions in a more intelligent way, ie, the rule : if - g = Fix g/1 := [y1:T1]...[yp:Tp]e & - f is guarded with respect to the set of pattern variables S in a1 ... am & - f is guarded with respect to the set of pattern variables S in T1 ... Tp & - ap is a sub-term of the formal argument of f & - f is guarded with respect to the set of pattern variables S+{yp} in e then f is guarded with respect to S in (g a1 ... am). Eduardo 7/9/98 *) | Fix ((recindxs,i),(_,typarray,bodies as recdef)) -> List.for_all (check_rec_call renv) l && array_for_all (check_rec_call renv) typarray && let decrArg = recindxs.(i) in let renv' = push_fix_renv renv recdef in if (List.length l < (decrArg+1)) then array_for_all (check_rec_call renv') bodies else let ok_vect = Array.mapi (fun j body -> if i=j then let decrarg_ind = inductive_of_fix renv'.env decrArg body in let theDecrArg = List.nth l decrArg in let arg_spec = subterm_specif renv theDecrArg decrarg_ind in check_nested_fix_body renv' (decrArg+1) arg_spec body else check_rec_call renv' body) bodies in array_for_all (fun b -> b) ok_vect | Const kn -> (try List.for_all (check_rec_call renv) l with (FixGuardError _ ) as e -> if evaluable_constant kn renv.env then check_rec_call renv (applist(constant_value renv.env kn, l)) else raise e) (* The cases below simply check recursively the condition on the subterms *) | Cast (a,_, b) -> List.for_all (check_rec_call renv) (a::b::l) | Lambda (x,a,b) -> check_rec_call (push_var_renv renv (x,a)) b && List.for_all (check_rec_call renv) (a::l) | Prod (x,a,b) -> check_rec_call (push_var_renv renv (x,a)) b && List.for_all (check_rec_call renv) (a::l) | CoFix (i,(_,typarray,bodies as recdef)) -> array_for_all (check_rec_call renv) typarray && List.for_all (check_rec_call renv) l && let renv' = push_fix_renv renv recdef in array_for_all (check_rec_call renv') bodies | Evar (_,la) -> array_for_all (check_rec_call renv) la && List.for_all (check_rec_call renv) l | Meta _ -> true | (App _ | LetIn _) -> anomaly "check_rec_call: should have been reduced" | (Ind _ | Construct _ | Var _ | Sort _) -> List.for_all (check_rec_call renv) l and check_nested_fix_body renv decr recArgsDecrArg body = if decr = 0 then check_rec_call (assign_var_spec renv (1,recArgsDecrArg)) body else match kind_of_term body with | Lambda (x,a,b) -> let renv' = push_var_renv renv (x,a) in check_rec_call renv a && check_nested_fix_body renv' (decr-1) recArgsDecrArg b | _ -> anomaly "Not enough abstractions in fix body" in check_rec_call renv def let inductive_of_mutfix env ((nvect,bodynum),(names,types,bodies as recdef)) = let nbfix = Array.length bodies in if nbfix = 0 or Array.length nvect <> nbfix or Array.length types <> nbfix or Array.length names <> nbfix or bodynum < 0 or bodynum >= nbfix then anomaly "Ill-formed fix term"; let fixenv = push_rec_types recdef env in let raise_err env i err = error_ill_formed_rec_body env err names i in (* Check the i-th definition with recarg k *) let find_ind i k def = if k < 0 then anomaly "negative recarg position"; (* check fi does not appear in the k+1 first abstractions, gives the type of the k+1-eme abstraction (must be an inductive) *) let rec check_occur env n def = match kind_of_term (whd_betadeltaiota env def) with | Lambda (x,a,b) -> if noccur_with_meta n nbfix a then let env' = push_rel (x, None, a) env in if n = k+1 then (* get the inductive type of the fixpoint *) let (mind, _) = try find_inductive env a with Not_found -> raise_err env i (RecursionNotOnInductiveType a) in (mind, (env', b)) else check_occur env' (n+1) b else anomaly "check_one_fix: Bad occurrence of recursive call" | _ -> raise_err env i NotEnoughAbstractionInFixBody in check_occur fixenv 1 def in (* Do it on every fixpoint *) let rv = array_map2_i find_ind nvect bodies in (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 for i = 0 to Array.length bodies - 1 do let (fenv,body) = rdef.(i) in let renv = make_renv fenv minds nvect.(i) minds.(i) in try let _ = check_one_fix renv nvect body in () with FixGuardError (fixenv,err) -> error_ill_formed_rec_body fixenv err names i done (* let cfkey = Profile.declare_profile "check_fix";; let check_fix env fix = Profile.profile3 cfkey check_fix env fix;; *) (************************************************************************) (* Scrape *) let rec scrape_mind env kn = match (Environ.lookup_mind kn env).mind_equiv with | None -> kn | Some kn' -> scrape_mind env kn' (************************************************************************) (* Co-fixpoints. *) exception CoFixGuardError of env * guard_error let anomaly_ill_typed () = anomaly "check_one_cofix: too many arguments applied to constructor" let rec codomain_is_coind env c = let b = whd_betadeltaiota env c in match kind_of_term b with | Prod (x,a,b) -> codomain_is_coind (push_rel (x, None, a) env) b | _ -> (try find_coinductive env b with Not_found -> raise (CoFixGuardError (env, CodomainNotInductiveType b))) let check_one_cofix env nbfix def deftype = let rec check_rec_call env alreadygrd n vlra t = if noccur_with_meta n nbfix t then true else let c,args = decompose_app (whd_betadeltaiota env t) in match kind_of_term c with | Meta _ -> true | Rel p when n <= p && p < n+nbfix -> (* recursive call *) if alreadygrd then if List.for_all (noccur_with_meta n nbfix) args then true else raise (CoFixGuardError (env,NestedRecursiveOccurrences)) else raise (CoFixGuardError (env,UnguardedRecursiveCall t)) | Construct (_,i as cstr_kn) -> let lra =vlra.(i-1) in let mI = inductive_of_constructor cstr_kn in let (mib,mip) = lookup_mind_specif env mI in let realargs = list_skipn mib.mind_nparams args in let rec process_args_of_constr = function | (t::lr), (rar::lrar) -> if rar = mk_norec then if noccur_with_meta n nbfix t then process_args_of_constr (lr, lrar) else raise (CoFixGuardError (env,RecCallInNonRecArgOfConstructor t)) else let spec = dest_subterms rar in check_rec_call env true n spec t && process_args_of_constr (lr, lrar) | [],_ -> true | _ -> anomaly_ill_typed () in process_args_of_constr (realargs, lra) | Lambda (x,a,b) -> assert (args = []); if (noccur_with_meta n nbfix a) then check_rec_call (push_rel (x, None, a) env) alreadygrd (n+1) vlra b else raise (CoFixGuardError (env,RecCallInTypeOfAbstraction a)) | CoFix (j,(_,varit,vdefs as recdef)) -> if (List.for_all (noccur_with_meta n nbfix) args) then let nbfix = Array.length vdefs in if (array_for_all (noccur_with_meta n nbfix) varit) then let env' = push_rec_types recdef env in (array_for_all (check_rec_call env' alreadygrd (n+1) vlra) vdefs) && (List.for_all (check_rec_call env alreadygrd (n+1) vlra) args) else raise (CoFixGuardError (env,RecCallInTypeOfDef c)) else raise (CoFixGuardError (env,UnguardedRecursiveCall c)) | Case (_,p,tm,vrest) -> if (noccur_with_meta n nbfix p) then if (noccur_with_meta n nbfix tm) then if (List.for_all (noccur_with_meta n nbfix) args) then (array_for_all (check_rec_call env alreadygrd n vlra) vrest) else raise (CoFixGuardError (env,RecCallInCaseFun c)) else raise (CoFixGuardError (env,RecCallInCaseArg c)) else raise (CoFixGuardError (env,RecCallInCasePred c)) | _ -> raise (CoFixGuardError (env,NotGuardedForm t)) in let (mind, _) = codomain_is_coind env deftype in let vlra = lookup_subterms env mind in check_rec_call env false 1 (dest_subterms vlra) def (* 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 let _ = check_one_cofix fixenv nbfix bodies.(i) types.(i) in () with CoFixGuardError (errenv,err) -> error_ill_formed_rec_body errenv err names i done