open Util open Names open Term open Pp open Indfun_common open Libnames open Rawterm open Declarations let is_rec_info scheme_info = let test_branche min acc (_,_,br) = acc || ( let new_branche = Sign.it_mkProd_or_LetIn mkProp (fst (Sign.decompose_prod_assum br)) in let free_rels_in_br = Termops.free_rels new_branche in let max = min + scheme_info.Tactics.npredicates in Util.Intset.exists (fun i -> i >= min && i< max) free_rels_in_br ) in Util.list_fold_left_i test_branche 1 false (List.rev scheme_info.Tactics.branches) let choose_dest_or_ind scheme_info = if is_rec_info scheme_info then Tactics.new_induct false else Tactics.new_destruct false let functional_induction with_clean c princl pat = let f,args = decompose_app c in fun g -> let princ,bindings, princ_type = match princl with | None -> (* No principle is given let's find the good one *) begin match kind_of_term f with | Const c' -> let princ_option = let finfo = (* we first try to find out a graph on f *) try find_Function_infos c' with Not_found -> errorlabstrm "" (str "Cannot find induction information on "++ Printer.pr_lconstr (mkConst c') ) in match Tacticals.elimination_sort_of_goal g with | InProp -> finfo.prop_lemma | InSet -> finfo.rec_lemma | InType -> finfo.rect_lemma in let princ = (* then we get the principle *) try mkConst (Option.get princ_option ) with Option.IsNone -> (*i If there is not default lemma defined then, we cross our finger and try to find a lemma named f_ind (or f_rec, f_rect) i*) let princ_name = Indrec.make_elimination_ident (id_of_label (con_label c')) (Tacticals.elimination_sort_of_goal g) in try mkConst(const_of_id princ_name ) with Not_found -> (* This one is neither defined ! *) errorlabstrm "" (str "Cannot find induction principle for " ++Printer.pr_lconstr (mkConst c') ) in (princ,Rawterm.NoBindings, Tacmach.pf_type_of g princ) | _ -> raise (UserError("",str "functional induction must be used with a function" )) end | Some ((princ,binding)) -> princ,binding,Tacmach.pf_type_of g princ in let princ_infos = Tactics.compute_elim_sig princ_type in let args_as_induction_constr = let c_list = if princ_infos.Tactics.farg_in_concl then [c] else [] in List.map (fun c -> Tacexpr.ElimOnConstr (c,NoBindings)) (args@c_list) in let princ' = Some (princ,bindings) in let princ_vars = List.fold_right (fun a acc -> try Idset.add (destVar a) acc with _ -> acc ) args Idset.empty in let old_idl = List.fold_right Idset.add (Tacmach.pf_ids_of_hyps g) Idset.empty in let old_idl = Idset.diff old_idl princ_vars in let subst_and_reduce g = if with_clean then let idl = map_succeed (fun id -> if Idset.mem id old_idl then failwith "subst_and_reduce"; id ) (Tacmach.pf_ids_of_hyps g) in let flag = Rawterm.Cbv {Rawterm.all_flags with Rawterm.rDelta = false; } in Tacticals.tclTHEN (Tacticals.tclMAP (fun id -> Tacticals.tclTRY (Equality.subst [id])) idl ) (Hiddentac.h_reduce flag Tacticals.allClauses) g else Tacticals.tclIDTAC g in Tacticals.tclTHEN (choose_dest_or_ind princ_infos args_as_induction_constr princ' pat) subst_and_reduce g type annot = Struct of identifier | Wf of Topconstr.constr_expr * identifier option * Topconstr.constr_expr list | Mes of Topconstr.constr_expr * identifier option * Topconstr.constr_expr list type newfixpoint_expr = identifier * annot * Topconstr.local_binder list * Topconstr.constr_expr * Topconstr.constr_expr let rec abstract_rawconstr c = function | [] -> c | Topconstr.LocalRawDef (x,b)::bl -> Topconstr.mkLetInC(x,b,abstract_rawconstr c bl) | Topconstr.LocalRawAssum (idl,k,t)::bl -> List.fold_right (fun x b -> Topconstr.mkLambdaC([x],k,t,b)) idl (abstract_rawconstr c bl) let interp_casted_constr_with_implicits sigma env impls c = (* Constrintern.interp_rawconstr_with_implicits sigma env [] impls c *) Constrintern.intern_gen false sigma env ~impls:([],impls) ~allow_patvar:false ~ltacvars:([],[]) c (* Construct a fixpoint as a Rawterm and not as a constr *) let build_newrecursive (lnameargsardef) = let env0 = Global.env() and sigma = Evd.empty in let (rec_sign,rec_impls) = List.fold_left (fun (env,impls) (recname,_,bl,arityc,_) -> let arityc = Command.generalize_constr_expr arityc bl in let arity = Constrintern.interp_type sigma env0 arityc in let impl = if Impargs.is_implicit_args() then Impargs.compute_implicits env0 arity else [] in let impls' =(recname,([],impl,Notation.compute_arguments_scope arity))::impls in (Environ.push_named (recname,None,arity) env, impls')) (env0,[]) lnameargsardef in let recdef = (* Declare local notations *) let fs = States.freeze() in let def = try List.map (fun (_,_,bl,_,def) -> let def = abstract_rawconstr def bl in interp_casted_constr_with_implicits sigma rec_sign rec_impls def ) lnameargsardef with e -> States.unfreeze fs; raise e in States.unfreeze fs; def in recdef,rec_impls let compute_annot (name,annot,args,types,body) = let names = List.map snd (Topconstr.names_of_local_assums args) in match annot with | None -> if List.length names > 1 then user_err_loc (dummy_loc,"Function", Pp.str "the recursive argument needs to be specified"); let new_annot = (id_of_name (List.hd names)) in (name,Struct new_annot,args,types,body) | Some r -> (name,r,args,types,body) (* Checks whether or not the mutual bloc is recursive *) let rec is_rec names = let names = List.fold_right Idset.add names Idset.empty in let check_id id names = Idset.mem id names in let rec lookup names = function | RVar(_,id) -> check_id id names | RRef _ | REvar _ | RPatVar _ | RSort _ | RHole _ | RDynamic _ -> false | RCast(_,b,_) -> lookup names b | RRec _ -> error "RRec not handled" | RIf(_,b,_,lhs,rhs) -> (lookup names b) || (lookup names lhs) || (lookup names rhs) | RLetIn(_,na,t,b) | RLambda(_,na,_,t,b) | RProd(_,na,_,t,b) -> lookup names t || lookup (Nameops.name_fold Idset.remove na names) b | RLetTuple(_,nal,_,t,b) -> lookup names t || lookup (List.fold_left (fun acc na -> Nameops.name_fold Idset.remove na acc) names nal ) b | RApp(_,f,args) -> List.exists (lookup names) (f::args) | RCases(_,_,el,brl) -> List.exists (fun (e,_) -> lookup names e) el || List.exists (lookup_br names) brl | RLetPattern (_,(c,_),p) -> lookup names c || lookup_br names p and lookup_br names (_,idl,_,rt) = let new_names = List.fold_right Idset.remove idl names in lookup new_names rt in lookup names let prepare_body (name,annot,args,types,body) rt = let n = (Topconstr.local_binders_length args) in (* Pp.msgnl (str "nb lambda to chop : " ++ str (string_of_int n) ++ fnl () ++Printer.pr_rawconstr rt); *) let fun_args,rt' = chop_rlambda_n n rt in (fun_args,rt') let derive_inversion fix_names = try (* we first transform the fix_names identifier into their corresponding constant *) let fix_names_as_constant = List.map (fun id -> destConst (Tacinterp.constr_of_id (Global.env ()) id)) fix_names in (* Then we check that the graphs have been defined If one of the graphs haven't been defined we do nothing *) List.iter (fun c -> ignore (find_Function_infos c)) fix_names_as_constant ; try Invfun.derive_correctness Functional_principles_types.make_scheme functional_induction fix_names_as_constant (*i The next call to mk_rel_id is valid since we have just construct the graph Ensures by : register_built i*) (List.map (fun id -> destInd (Tacinterp.constr_of_id (Global.env ()) (mk_rel_id id))) fix_names ) with e -> msg_warning (str "Cannot built inversion information" ++ if do_observe () then Cerrors.explain_exn e else mt ()) with _ -> () let warning_error names e = match e with | Building_graph e -> Pp.msg_warning (str "Cannot define graph(s) for " ++ h 1 (prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names) ++ if do_observe () then (spc () ++ Cerrors.explain_exn e) else mt ()) | Defining_principle e -> Pp.msg_warning (str "Cannot define principle(s) for "++ h 1 (prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names) ++ if do_observe () then Cerrors.explain_exn e else mt ()) | _ -> anomaly "" let error_error names e = match e with | Building_graph e -> errorlabstrm "" (str "Cannot define graph(s) for " ++ h 1 (prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names) ++ if do_observe () then (spc () ++ Cerrors.explain_exn e) else mt ()) | _ -> anomaly "" let generate_principle on_error is_general do_built fix_rec_l recdefs interactive_proof (continue_proof : int -> Names.constant array -> Term.constr array -> int -> Tacmach.tactic) : unit = let names = List.map (function (name,_,_,_,_) -> name) fix_rec_l in let fun_bodies = List.map2 prepare_body fix_rec_l recdefs in let funs_args = List.map fst fun_bodies in let funs_types = List.map (function (_,_,_,types,_) -> types) fix_rec_l in try (* We then register the Inductive graphs of the functions *) Rawterm_to_relation.build_inductive names funs_args funs_types recdefs; if do_built then begin (*i The next call to mk_rel_id is valid since we have just construct the graph Ensures by : do_built i*) let f_R_mut = Ident (dummy_loc,mk_rel_id (List.nth names 0)) in let ind_kn = fst (locate_with_msg (pr_reference f_R_mut++str ": Not an inductive type!") locate_ind f_R_mut) in let fname_kn (fname,_,_,_,_) = let f_ref = Ident (dummy_loc,fname) in locate_with_msg (pr_reference f_ref++str ": Not an inductive type!") locate_constant f_ref in let funs_kn = Array.of_list (List.map fname_kn fix_rec_l) in let _ = list_map_i (fun i x -> let princ = destConst (Indrec.lookup_eliminator (ind_kn,i) (InProp)) in let princ_type = Typeops.type_of_constant (Global.env()) princ in Functional_principles_types.generate_functional_principle interactive_proof princ_type None None funs_kn i (continue_proof 0 [|funs_kn.(i)|]) ) 0 fix_rec_l in Array.iter (add_Function is_general) funs_kn; () end with e -> on_error names e let register_struct is_rec fixpoint_exprl = match fixpoint_exprl with | [(fname,_,bl,ret_type,body),_] when not is_rec -> Command.declare_definition fname (Decl_kinds.Global,Flags.boxed_definitions (),Decl_kinds.Definition) bl None body (Some ret_type) (fun _ _ -> ()) | _ -> Command.build_recursive fixpoint_exprl (Flags.boxed_definitions()) let generate_correction_proof_wf f_ref tcc_lemma_ref is_mes functional_ref eq_ref rec_arg_num rec_arg_type nb_args relation (_: int) (_:Names.constant array) (_:Term.constr array) (_:int) : Tacmach.tactic = Functional_principles_proofs.prove_principle_for_gen (f_ref,functional_ref,eq_ref) tcc_lemma_ref is_mes rec_arg_num rec_arg_type relation let register_wf ?(is_mes=false) fname rec_impls wf_rel_expr wf_arg using_lemmas args ret_type body pre_hook = let type_of_f = Command.generalize_constr_expr ret_type args in let rec_arg_num = let names = List.map snd (Topconstr.names_of_local_assums args) in match wf_arg with | None -> if List.length names = 1 then 1 else error "Recursive argument must be specified" | Some wf_arg -> list_index (Name wf_arg) names in let unbounded_eq = let f_app_args = Topconstr.CAppExpl (dummy_loc, (None,(Ident (dummy_loc,fname))) , (List.map (function | _,Anonymous -> assert false | _,Name e -> (Topconstr.mkIdentC e) ) (Topconstr.names_of_local_assums args) ) ) in Topconstr.CApp (dummy_loc,(None,Topconstr.mkIdentC (id_of_string "eq")), [(f_app_args,None);(body,None)]) in let eq = Command.generalize_constr_expr unbounded_eq args in let hook f_ref tcc_lemma_ref functional_ref eq_ref rec_arg_num rec_arg_type nb_args relation = try pre_hook (generate_correction_proof_wf f_ref tcc_lemma_ref is_mes functional_ref eq_ref rec_arg_num rec_arg_type nb_args relation ); derive_inversion [fname] with e -> (* No proof done *) () in Recdef.recursive_definition is_mes fname rec_impls type_of_f wf_rel_expr rec_arg_num eq hook using_lemmas let register_mes fname rec_impls wf_mes_expr wf_arg using_lemmas args ret_type body = let wf_arg_type,wf_arg = match wf_arg with | None -> begin match args with | [Topconstr.LocalRawAssum ([(_,Name x)],k,t)] -> t,x | _ -> error "Recursive argument must be specified" end | Some wf_args -> try match List.find (function | Topconstr.LocalRawAssum(l,k,t) -> List.exists (function (_,Name id) -> id = wf_args | _ -> false) l | _ -> false ) args with | Topconstr.LocalRawAssum(_,k,t) -> t,wf_args | _ -> assert false with Not_found -> assert false in let ltof = let make_dir l = make_dirpath (List.map id_of_string (List.rev l)) in Libnames.Qualid (dummy_loc,Libnames.qualid_of_sp (Libnames.make_path (make_dir ["Arith";"Wf_nat"]) (id_of_string "ltof"))) in let fun_from_mes = let applied_mes = Topconstr.mkAppC(wf_mes_expr,[Topconstr.mkIdentC wf_arg]) in Topconstr.mkLambdaC ([(dummy_loc,Name wf_arg)],Topconstr.default_binder_kind,wf_arg_type,applied_mes) in let wf_rel_from_mes = Topconstr.mkAppC(Topconstr.mkRefC ltof,[wf_arg_type;fun_from_mes]) in register_wf ~is_mes:true fname rec_impls wf_rel_from_mes (Some wf_arg) using_lemmas args ret_type body let do_generate_principle on_error register_built interactive_proof fixpoint_exprl = let recdefs,rec_impls = build_newrecursive fixpoint_exprl in let _is_struct = match fixpoint_exprl with | [((name,Some (Wf (wf_rel,wf_x,using_lemmas)),args,types,body))] -> let pre_hook = generate_principle on_error true register_built fixpoint_exprl recdefs true in if register_built then register_wf name rec_impls wf_rel wf_x using_lemmas args types body pre_hook; false | [((name,Some (Mes (wf_mes,wf_x,using_lemmas)),args,types,body))] -> let pre_hook = generate_principle on_error true register_built fixpoint_exprl recdefs true in if register_built then register_mes name rec_impls wf_mes wf_x using_lemmas args types body pre_hook; true | _ -> let fix_names = List.map (function (name,_,_,_,_) -> name) fixpoint_exprl in let is_one_rec = is_rec fix_names in let old_fixpoint_exprl = List.map (function | (name,Some (Struct id),args,types,body),_ -> let names = List.map snd (Topconstr.names_of_local_assums args) in let annot = try Some (list_index0 (Name id) names), Topconstr.CStructRec with Not_found -> raise (UserError("",str "Cannot find argument " ++ Ppconstr.pr_id id)) in (name,annot,args,types,body),(None:Vernacexpr.decl_notation) | (name,None,args,types,body),recdef -> let names = (Topconstr.names_of_local_assums args) in if is_one_rec recdef && List.length names > 1 then user_err_loc (dummy_loc,"Function", Pp.str "the recursive argument needs to be specified in Function") else (name,(Some 0, Topconstr.CStructRec),args,types,body), (None:Vernacexpr.decl_notation) | (_,Some (Wf _),_,_,_),_ | (_,Some (Mes _),_,_,_),_-> error ("Cannot use mutual definition with well-founded recursion or measure") ) (List.combine fixpoint_exprl recdefs) in (* ok all the expressions are structural *) let fix_names = List.map (function (name,_,_,_,_) -> name) fixpoint_exprl in let is_rec = List.exists (is_rec fix_names) recdefs in if register_built then register_struct is_rec old_fixpoint_exprl; generate_principle on_error false register_built fixpoint_exprl recdefs interactive_proof (Functional_principles_proofs.prove_princ_for_struct interactive_proof); if register_built then derive_inversion fix_names; true; in () open Topconstr let rec add_args id new_args b = match b with | CRef r -> begin match r with | Libnames.Ident(loc,fname) when fname = id -> CAppExpl(dummy_loc,(None,r),new_args) | _ -> b end | CFix _ | CCoFix _ -> anomaly "add_args : todo" | CArrow(loc,b1,b2) -> CArrow(loc,add_args id new_args b1, add_args id new_args b2) | CProdN(loc,nal,b1) -> CProdN(loc, List.map (fun (nal,k,b2) -> (nal,k,add_args id new_args b2)) nal, add_args id new_args b1) | CLambdaN(loc,nal,b1) -> CLambdaN(loc, List.map (fun (nal,k,b2) -> (nal,k,add_args id new_args b2)) nal, add_args id new_args b1) | CLetIn(loc,na,b1,b2) -> CLetIn(loc,na,add_args id new_args b1,add_args id new_args b2) | CAppExpl(loc,(pf,r),exprl) -> begin match r with | Libnames.Ident(loc,fname) when fname = id -> CAppExpl(loc,(pf,r),new_args@(List.map (add_args id new_args) exprl)) | _ -> CAppExpl(loc,(pf,r),List.map (add_args id new_args) exprl) end | CApp(loc,(pf,b),bl) -> CApp(loc,(pf,add_args id new_args b), List.map (fun (e,o) -> add_args id new_args e,o) bl) | CCases(loc,b_option,cel,cal) -> CCases(loc,Option.map (add_args id new_args) b_option, List.map (fun (b,(na,b_option)) -> add_args id new_args b, (na,Option.map (add_args id new_args) b_option)) cel, List.map (fun (loc,cpl,e) -> (loc,cpl,add_args id new_args e)) cal ) | CLetTuple(loc,nal,(na,b_option),b1,b2) -> CLetTuple(loc,nal,(na,Option.map (add_args id new_args) b_option), add_args id new_args b1, add_args id new_args b2 ) | CLetPattern(loc,p,b,c) -> CLetPattern(loc,p,add_args id new_args b,add_args id new_args c) | CIf(loc,b1,(na,b_option),b2,b3) -> CIf(loc,add_args id new_args b1, (na,Option.map (add_args id new_args) b_option), add_args id new_args b2, add_args id new_args b3 ) | CHole _ -> b | CPatVar _ -> b | CEvar _ -> b | CSort _ -> b | CCast(loc,b1,CastConv(ck,b2)) -> CCast(loc,add_args id new_args b1,CastConv(ck,add_args id new_args b2)) | CCast(loc,b1,CastCoerce) -> CCast(loc,add_args id new_args b1,CastCoerce) | CNotation _ -> anomaly "add_args : CNotation" | CPrim _ -> b | CDelimiters _ -> anomaly "add_args : CDelimiters" | CDynamic _ -> anomaly "add_args : CDynamic" exception Stop of Topconstr.constr_expr (* [chop_n_arrow n t] chops the [n] first arrows in [t] Acts on Topconstr.constr_expr *) let rec chop_n_arrow n t = if n <= 0 then t (* If we have already removed all the arrows then return the type *) else (* If not we check the form of [t] *) match t with | Topconstr.CArrow(_,_,t) -> (* If we have an arrow, we discard it and recall [chop_n_arrow] *) chop_n_arrow (n-1) t | Topconstr.CProdN(_,nal_ta',t') -> (* If we have a forall, to result are possible : either we need to discard more than the number of arrows contained in this product declaration then we just recall [chop_n_arrow] on the remaining number of arrow to chop and [t'] we discard it and recall [chop_n_arrow], either this product contains more arrows than the number we need to chop and then we return the new type *) begin try let new_n = let rec aux (n:int) = function [] -> n | (nal,k,t'')::nal_ta' -> let nal_l = List.length nal in if n >= nal_l then aux (n - nal_l) nal_ta' else let new_t' = Topconstr.CProdN(dummy_loc, ((snd (list_chop n nal)),k,t'')::nal_ta',t') in raise (Stop new_t') in aux n nal_ta' in chop_n_arrow new_n t' with Stop t -> t end | _ -> anomaly "Not enough products" let rec get_args b t : Topconstr.local_binder list * Topconstr.constr_expr * Topconstr.constr_expr = match b with | Topconstr.CLambdaN (loc, (nal_ta), b') -> begin let n = (List.fold_left (fun n (nal,_,_) -> n+List.length nal) 0 nal_ta ) in let nal_tas,b'',t'' = get_args b' (chop_n_arrow n t) in (List.map (fun (nal,k,ta) -> (Topconstr.LocalRawAssum (nal,k,ta))) nal_ta)@nal_tas, b'',t'' end | _ -> [],b,t let make_graph (f_ref:global_reference) = let c,c_body = match f_ref with | ConstRef c -> begin try c,Global.lookup_constant c with Not_found -> raise (UserError ("",str "Cannot find " ++ Printer.pr_lconstr (mkConst c)) ) end | _ -> raise (UserError ("", str "Not a function reference") ) in match c_body.const_body with | None -> error "Cannot build a graph over an axiom !" | Some b -> let env = Global.env () in let body = (force b) in let extern_body,extern_type = with_full_print (fun () -> (Constrextern.extern_constr false env body, Constrextern.extern_type false env (Typeops.type_of_constant_type env c_body.const_type) ) ) () in let (nal_tas,b,t) = get_args extern_body extern_type in let expr_list = match b with | Topconstr.CFix(loc,l_id,fixexprl) -> let l = List.map (fun (id,(n,recexp),bl,t,b) -> let bl' = List.flatten (List.map (function | Topconstr.LocalRawDef (na,_)-> [] | Topconstr.LocalRawAssum (nal,_,_) -> nal ) bl ) in let rec_id = match List.nth bl' (Option.get n) with |(_,Name id) -> id | _ -> anomaly "" in let new_args = List.flatten (List.map (function | Topconstr.LocalRawDef (na,_)-> [] | Topconstr.LocalRawAssum (nal,_,_) -> List.map (fun (loc,n) -> CRef(Libnames.Ident(loc, Nameops.out_name n))) nal ) nal_tas ) in let b' = add_args id new_args b in (id, Some (Struct rec_id),nal_tas@bl,t,b') ) fixexprl in l | _ -> let id = id_of_label (con_label c) in [(id,None,nal_tas,t,b)] in do_generate_principle error_error false false expr_list; (* We register the infos *) let mp,dp,_ = repr_con c in List.iter (fun (id,_,_,_,_) -> add_Function false (make_con mp dp (label_of_id id))) expr_list (* let make_graph _ = assert false *) let do_generate_principle = do_generate_principle warning_error true