(************************************************************************) (* 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 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 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 t with | Ind ind when not (fst (lookup_mind_specif env ind)).mind_finite -> (ind, l) | _ -> raise Not_found let inductive_params (mib,_) = mib.mind_nparams (************************************************************************) (* 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 = Ind (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) = fold_rel_context (fun (_,copt,_) (largs,subs,ty) -> match (copt, largs, 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 full_inductive_instantiate mib params sign = let dummy = Prop Null in let t = mkArity (sign,dummy) in fst (destArity (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 *) (* 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 sort_as_univ = function | Type u -> u | Prop Null -> type0m_univ | Prop Pos -> type0_univ let cons_subst u su subst = try (u, sup su (List.assoc u subst)) :: List.remove_assoc u subst with Not_found -> (u, su) :: subst let actualize_decl_level env lev t = let sign,s = dest_arity env t in mkArity (sign,lev) let polymorphism_on_non_applied_parameters = false (* Bind expected levels of parameters to actual levels *) (* Propagate the new levels in the signature *) let rec make_subst env = function | (_,Some _,_ as t)::sign, exp, args -> let ctx,subst = make_subst env (sign, exp, args) in t::ctx, subst | d::sign, None::exp, args -> let args = match args with _::args -> args | [] -> [] in let ctx,subst = make_subst env (sign, exp, args) in d::ctx, subst | d::sign, Some u::exp, a::args -> (* We recover the level of the argument, but we don't change the *) (* level in the corresponding type in the arity; this level in the *) (* arity is a global level which, at typing time, will be enforce *) (* to be greater than the level of the argument; this is probably *) (* a useless extra constraint *) let s = sort_as_univ (snd (dest_arity env a)) in let ctx,subst = make_subst env (sign, exp, args) in d::ctx, cons_subst u s subst | (na,None,t as d)::sign, Some u::exp, [] -> (* No more argument here: we instantiate the type with a fresh level *) (* which is first propagated to the corresponding premise in the arity *) (* (actualize_decl_level), then to the conclusion of the arity (via *) (* the substitution) *) let ctx,subst = make_subst env (sign, exp, []) in if polymorphism_on_non_applied_parameters then let s = fresh_local_univ () in let t = actualize_decl_level env (Type s) t in (na,None,t)::ctx, cons_subst u s subst else d::ctx, subst | sign, [], _ -> (* Uniform parameters are exhausted *) sign,[] | [], _, _ -> assert false let instantiate_universes env ctx ar argsorts = let args = Array.to_list argsorts in let ctx,subst = make_subst env (ctx,ar.poly_param_levels,args) in let level = subst_large_constraints subst ar.poly_level in ctx, if is_type0m_univ level then Prop Null else if is_type0_univ level then Prop Pos else Type level let type_of_inductive_knowing_parameters env mip paramtyps = match mip.mind_arity with | Monomorphic s -> s.mind_user_arity | Polymorphic ar -> let ctx = List.rev mip.mind_arity_ctxt in let ctx,s = instantiate_universes env ctx ar paramtyps in mkArity (List.rev ctx,s) (* Type of a (non applied) inductive type *) let type_of_inductive env (_,mip) = type_of_inductive_knowing_parameters env mip [||] (* The max of an array of universes *) let cumulate_constructor_univ u = function | Prop Null -> u | Prop Pos -> sup type0_univ u | Type u' -> sup u u' let max_inductive_sort = Array.fold_left cumulate_constructor_univ type0m_univ (************************************************************************) (* 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 error_elim_expln kp ki = match kp,ki with | (InType | InSet), InProp -> NonInformativeToInformative | InType, InSet -> StrongEliminationOnNonSmallType (* if Set impredicative *) | _ -> WrongArity (* Type of case predicates *) (* Get type of inductive, with parameters instantiated *) let inductive_sort_family mip = match mip.mind_arity with | Monomorphic s -> family_of_sort s.mind_sort | Polymorphic _ -> InType let mind_arity mip = mip.mind_arity_ctxt, inductive_sort_family mip let get_instantiated_arity (mib,mip) params = let sign, s = mind_arity mip in full_inductive_instantiate mib params sign, s let elim_sorts (_,mip) = mip.mind_kelim let extended_rel_list n hyps = let rec reln l p = function | (_,None,_) :: hyps -> reln (Rel (n+p) :: l) (p+1) hyps | (_,Some _,_) :: hyps -> reln l (p+1) hyps | [] -> l in reln [] 1 hyps let build_dependent_inductive ind (_,mip) params = let realargs,_ = list_chop mip.mind_nrealargs_ctxt mip.mind_arity_ctxt in applist (Ind ind, List.map (lift mip.mind_nrealargs_ctxt) params @ extended_rel_list 0 realargs) (* This exception is local *) exception LocalArity of (sorts_family * sorts_family * arity_error) option let check_allowed_sort ksort specif = if not (List.exists ((=) ksort) (elim_sorts specif)) then let s = inductive_sort_family (snd specif) in raise (LocalArity (Some(ksort,s,error_elim_expln ksort s))) let is_correct_arity env c (p,pj) ind specif params = let arsign,_ = get_instantiated_arity specif params in let rec srec env pt ar = let pt' = whd_betadeltaiota env pt in match pt', ar with | Prod (na1,a1,t), (_,None,a1')::ar' -> (try conv env a1 a1' with NotConvertible -> raise (LocalArity None)); srec (push_rel (na1,None,a1) env) t ar' | Prod (na1,a1,a2), [] -> (* whnf of t was not needed here! *) let env' = push_rel (na1,None,a1) env in let ksort = match (whd_betadeltaiota env' a2) with | Sort s -> family_of_sort s | _ -> raise (LocalArity None) in let dep_ind = build_dependent_inductive ind specif params in (try conv env a1 dep_ind with NotConvertible -> raise (LocalArity None)); check_allowed_sort ksort specif; true | Sort s', [] -> check_allowed_sort (family_of_sort s') specif; false | _, (_,Some _,_ as d)::ar' -> srec (push_rel d env) (lift 1 pt') ar' | _ -> raise (LocalArity None) in try srec env pj (List.rev arsign) with LocalArity kinds -> error_elim_arity env ind (elim_sorts specif) c (p,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 (_,mip as specif) params dep p = let build_one_branch i cty = let typi = full_constructor_instantiate (ind,specif,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 (inductive_params specif) allargs in let cargs = if dep then let cstr = ith_constructor_of_inductive ind (i+1) in let dep_cstr = applist (Construct cstr,lparams@extended_rel_list 0 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) (p,pj) c = let specif = lookup_mind_specif env ind in let nparams = inductive_params specif in let (params,realargs) = list_chop nparams largs in let dep = is_correct_arity env c (p,pj) ind specif params in let lc = build_branches_type ind specif params dep p in let ty = build_case_type dep p c realargs in (lc, ty) (************************************************************************) (* Checking the case annotation is relevant *) let check_case_info env indsp ci = let (mib,mip) = lookup_mind_specif env indsp in if not (eq_ind indsp ci.ci_ind) or (mib.mind_nparams <> ci.ci_npar) or (mip.mind_consnrealdecls <> ci.ci_cstr_ndecls) 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 *) (* tells whether it is a strict or loose subterm *) type size = Large | Strict (* merging information *) let size_glb s1 s2 = match s1,s2 with Strict, Strict -> Strict | _ -> Large (* possible specifications for a term: - Not_subterm: when the size of a term is not related to the recursive argument of the fixpoint - Subterm: when the term is a subterm of the recursive argument the wf_paths argument specifies which subterms are recursive - Dead_code: when the term has been built by elimination over an empty type *) type subterm_spec = Subterm of (size * wf_paths) | Dead_code | Not_subterm let spec_of_tree t = lazy (if Rtree.eq_rtree (=) (Lazy.force t) mk_norec then Not_subterm else Subterm(Strict,Lazy.force 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 Rtree.eq_rtree (=) 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 Lazy.t list; } let make_renv env minds recarg (kn,tyi) = let mib = 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 = [Lazy.lazy_from_val(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,Lazy.lazy_from_val Not_subterm) (* Fetch recursive information about a variable p *) let subterm_var p renv = try Lazy.force (List.nth renv.genv (p-1)) with Failure _ | Invalid_argument _ -> Not_subterm 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 -> Lazy.lazy_from_val 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 -> Lazy.lazy_from_val Not_subterm::ge) n renv.genv } (* Definition and manipulation of the stack *) type stack_element = |SClosure of guard_env*constr |SArg of subterm_spec Lazy.t let push_stack_closures renv l stack = List.fold_right (fun h b -> (SClosure (renv,h))::b) l stack let push_stack_args l stack = List.fold_right (fun h b -> (SArg h)::b) l stack (******************************) (* 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 let match_inductive ind ra = match ra with | (Mrec i | Imbr i) -> eq_ind ind i | Norec -> false (* In {match c as z in ci y_s return P with |C_i x_s => t end} [branches_specif renv c_spec ci] returns an array of x_s specs knowing c_spec. *) let branches_specif renv c_spec ci = let car = (* We fetch the regular tree associated to the inductive of the match. This is just to get the number of constructors (and constructor arities) that fit the match branches without forcing c_spec. Note that c_spec might be more precise than [v] below, because of nested inductive types. *) let (_,mip) = lookup_mind_specif renv.env ci.ci_ind in let v = dest_subterms mip.mind_recargs in Array.map List.length v in Array.mapi (fun i nca -> (* i+1-th cstructor has arity nca *) let lvra = lazy (match Lazy.force c_spec with Subterm (_,t) when match_inductive ci.ci_ind (dest_recarg t) -> let vra = Array.of_list (dest_subterms t).(i) in assert (nca = Array.length vra); Array.map (fun t -> Lazy.force (spec_of_tree (lazy t))) vra | Dead_code -> Array.create nca Dead_code | _ -> Array.create nca Not_subterm) in list_tabulate (fun j -> lazy (Lazy.force lvra).(j)) nca) car (* [subterm_specif renv t] computes the recursive structure of [t] and compare its size with the size of the initial recursive argument of the fixpoint we are checking. [renv] collects such information about variables. *) let rec subterm_specif renv stack t = (* maybe reduction is not always necessary! *) let f,l = decompose_app (whd_betadeltaiota renv.env t) in match f with | Rel k -> subterm_var k renv | Case (ci,_,c,lbr) -> let stack' = push_stack_closures renv l stack in let cases_spec = branches_specif renv (lazy_subterm_specif renv [] c) ci in let stl = Array.mapi (fun i br' -> let stack_br = push_stack_args (cases_spec.(i)) stack' in subterm_specif renv stack_br br') lbr 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 (ctxt,clfix) = dest_prod renv.env typarray.(i) in let oind = let env' = push_rel_context ctxt renv.env in try Some(fst(find_inductive env' clfix)) with Not_found -> None in (match oind with None -> Not_subterm (* happens if fix is polymorphic *) | Some ind -> 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' = (* Why Strict here ? To be general, it could also be Large... *) assign_var_spec renv' (nbfix-i, lazy (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 stack' = push_stack_closures renv l stack in let renv'' = push_ctxt_renv renv' sign in let renv'' = if List.length stack' < nbOfAbst then renv'' else let decrArg = List.nth stack' decrArg in let arg_spec = stack_element_specif decrArg in assign_var_spec renv'' (1, arg_spec) in subterm_specif renv'' [] strippedBody) | Lambda (x,a,b) -> assert (l=[]); let spec,stack' = extract_stack renv a stack in subterm_specif (push_var renv (x,a,spec)) stack' b (* Metas and evars are considered OK *) | (Meta _|Evar _) -> Dead_code (* Other terms are not subterms *) | _ -> Not_subterm and lazy_subterm_specif renv stack t = lazy (subterm_specif renv stack t) and stack_element_specif = function |SClosure (h_renv,h) -> lazy_subterm_specif h_renv [] h |SArg x -> x and extract_stack renv a = function | [] -> Lazy.lazy_from_val Not_subterm , [] | h::t -> stack_element_specif h, t (* Check size x is a correct size for recursive calls. *) let check_is_subterm x = match Lazy.force x with Subterm (Strict,_) | Dead_code -> true | _ -> false (************************************************************************) exception FixGuardError of env * guard_error let error_illegal_rec_call renv fx (arg_renv,arg) = let (_,le_vars,lt_vars) = List.fold_left (fun (i,le,lt) sbt -> match Lazy.force 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_renv.env, 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 (* Checks if [t] only make valid recursive calls *) let rec check_rec_call renv stack t = (* if [t] does not make recursive calls, it is guarded: *) if noccur_with_meta renv.rel_min nfi t then () else let (f,l) = decompose_app (whd_betaiotazeta t) in match f with | Rel p -> (* Test if [p] is a fixpoint (recursive call) *) if renv.rel_min <= p & p < renv.rel_min+nfi then begin List.iter (check_rec_call renv []) l; (* 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 let stack' = push_stack_closures renv l stack in if List.length stack' <= np then error_partial_apply renv glob else (* Check the decreasing arg is smaller *) let z = List.nth stack' np in if not (check_is_subterm (stack_element_specif z)) then begin match z with |SClosure (z,z') -> error_illegal_rec_call renv glob (z,z') |SArg _ -> error_partial_apply renv glob end end else begin match pi2 (lookup_rel p renv.env) with | None -> List.iter (check_rec_call renv []) l | Some c -> try List.iter (check_rec_call renv []) l with FixGuardError _ -> check_rec_call renv stack (applist(lift p c,l)) end | Case (ci,p,c_0,lrest) -> List.iter (check_rec_call renv []) (c_0::p::l); (* compute the recarg information for the arguments of each branch *) let case_spec = branches_specif renv (lazy_subterm_specif renv [] c_0) ci in let stack' = push_stack_closures renv l stack in Array.iteri (fun k br' -> let stack_br = push_stack_args case_spec.(k) stack' in check_rec_call renv stack_br br') lrest (* Enables to traverse Fixpoint definitions in a more intelligent way, ie, the rule : if - g = fix g (y1:T1)...(yp:Tp) {struct yp} := 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.iter (check_rec_call renv []) l; Array.iter (check_rec_call renv []) typarray; let decrArg = recindxs.(i) in let renv' = push_fix_renv renv recdef in let stack' = push_stack_closures renv l stack in Array.iteri (fun j body -> if i=j && (List.length stack' > decrArg) then let recArg = List.nth stack' decrArg in let arg_sp = stack_element_specif recArg in check_nested_fix_body renv' (decrArg+1) arg_sp body else check_rec_call renv' [] body) bodies | Const kn -> if evaluable_constant kn renv.env then try List.iter (check_rec_call renv []) l with (FixGuardError _ ) -> let value = (applist(constant_value renv.env kn, l)) in check_rec_call renv stack value else List.iter (check_rec_call renv []) l | Lambda (x,a,b) -> assert (l = []); check_rec_call renv [] a ; let spec, stack' = extract_stack renv a stack in check_rec_call (push_var renv (x,a,spec)) stack' b | Prod (x,a,b) -> assert (l = [] && stack = []); check_rec_call renv [] a; check_rec_call (push_var_renv renv (x,a)) [] b | CoFix (i,(_,typarray,bodies as recdef)) -> List.iter (check_rec_call renv []) l; Array.iter (check_rec_call renv []) typarray; let renv' = push_fix_renv renv recdef in Array.iter (check_rec_call renv' []) bodies | (Ind _ | Construct _) -> List.iter (check_rec_call renv []) l | Var id -> begin match pi2 (lookup_named id renv.env) with | None -> List.iter (check_rec_call renv []) l | Some c -> try List.iter (check_rec_call renv []) l with (FixGuardError _) -> check_rec_call renv stack (applist(c,l)) end | Sort _ -> assert (l = []) (* l is not checked because it is considered as the meta's context *) | (Evar _ | Meta _) -> () | (App _ | LetIn _ | Cast _) -> assert false (* beta zeta reduction *) 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 body with | Lambda (x,a,b) -> check_rec_call renv [] a; let renv' = push_var_renv renv (x,a) in 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 = (* 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 (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 check_one_fix renv nvect body 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;; *) (************************************************************************) (* 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 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 not (noccur_with_meta n nbfix t) then let c,args = decompose_app (whd_betadeltaiota env t) in match c with | Rel p when n <= p && p < n+nbfix -> (* recursive call: must be guarded and no nested recursive call allowed *) if not alreadygrd then raise (CoFixGuardError (env,UnguardedRecursiveCall t)) else if not(List.for_all (noccur_with_meta n nbfix) args) then raise (CoFixGuardError (env,NestedRecursiveOccurrences)) | 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) | [],_ -> () | _ -> anomaly_ill_typed () in process_args_of_constr (realargs, lra) | Lambda (x,a,b) -> assert (args = []); if noccur_with_meta n nbfix a then let env' = push_rel (x, None, a) env in check_rec_call 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 if array_for_all (noccur_with_meta n nbfix) varit then let nbfix = Array.length vdefs in let env' = push_rec_types recdef env in (Array.iter (check_rec_call env' alreadygrd (n+nbfix) vlra) vdefs; List.iter (check_rec_call env alreadygrd n 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.iter (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)) | Meta _ -> () | Evar _ -> List.iter (check_rec_call env alreadygrd n vlra) args | _ -> 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 check_one_cofix fixenv nbfix bodies.(i) types.(i) with CoFixGuardError (errenv,err) -> error_ill_formed_rec_body errenv err names i done