(************************************************************************) (* * The Coq Proof Assistant / The Coq Development Team *) (* v * INRIA, CNRS and contributors - Copyright 1999-2019 *) (* assert false in try let v = tac g in msgnl (goal ++ fnl () ++ s ++(str " ")++(str "finished")); v with reraise -> let reraise = CErrors.push reraise in observe (hov 0 (str "observation "++ s++str " raised exception " ++ CErrors.iprint reraise ++ str " on goal" ++ fnl() ++ goal )); iraise reraise;; let observe_tac s tac g = if do_observe () then do_observe_tac (str s) tac g else tac g let is_rec_info sigma scheme_info = let test_branche min acc decl = acc || ( let new_branche = it_mkProd_or_LetIn mkProp (fst (decompose_prod_assum sigma (RelDecl.get_type decl))) in let free_rels_in_br = Termops.free_rels sigma new_branche in let max = min + scheme_info.Tactics.npredicates in Int.Set.exists (fun i -> i >= min && i< max) free_rels_in_br ) in List.fold_left_i test_branche 1 false (List.rev scheme_info.Tactics.branches) let choose_dest_or_ind scheme_info args = Proofview.tclBIND Proofview.tclEVARMAP (fun sigma -> Tactics.induction_destruct (is_rec_info sigma scheme_info) false args) let functional_induction with_clean c princl pat = let res = fun g -> let sigma = Tacmach.project g in let f,args = decompose_app sigma c in let princ,bindings, princ_type,g' = match princl with | None -> (* No principle is given let's find the good one *) begin match EConstr.kind sigma f with | Const (c',u) -> let princ_option = let finfo = (* we first try to find out a graph on f *) try find_Function_infos c' with Not_found -> user_err (str "Cannot find induction information on "++ Printer.pr_leconstr_env (Tacmach.pf_env g) sigma (mkConst c') ) in match Tacticals.elimination_sort_of_goal g with | InSProp -> finfo.sprop_lemma | InProp -> finfo.prop_lemma | InSet -> finfo.rec_lemma | InType -> finfo.rect_lemma in let princ,g' = (* then we get the principle *) try let g',princ = Tacmach.pf_eapply (Evd.fresh_global) g (GlobRef.ConstRef (Option.get princ_option )) in princ,g' 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 (Label.to_id (Constant.label c')) (Tacticals.elimination_sort_of_goal g) in try let princ_ref = const_of_id princ_name in let (a,b) = Tacmach.pf_eapply (Evd.fresh_global) g princ_ref in (b,a) (* mkConst(const_of_id princ_name ),g (\* FIXME *\) *) with Not_found -> (* This one is neither defined ! *) user_err (str "Cannot find induction principle for " ++ Printer.pr_leconstr_env (Tacmach.pf_env g) sigma (mkConst c') ) in (princ,NoBindings,Tacmach.pf_unsafe_type_of g' princ,g') | _ -> raise (UserError(None,str "functional induction must be used with a function" )) end | Some ((princ,binding)) -> princ,binding,Tacmach.pf_unsafe_type_of g princ,g in let sigma = Tacmach.project g' in let princ_infos = Tactics.compute_elim_sig (Tacmach.project g') princ_type in let args_as_induction_constr = let c_list = if princ_infos.Tactics.farg_in_concl then [c] else [] in if List.length args + List.length c_list = 0 then user_err Pp.(str "Cannot recognize a valid functional scheme" ); let encoded_pat_as_patlist = List.make (List.length args + List.length c_list - 1) None @ [pat] in List.map2 (fun c pat -> ((None, Tactics.ElimOnConstr (fun env sigma -> (sigma,(c,NoBindings)))), (None,pat), None)) (args@c_list) encoded_pat_as_patlist in let princ' = Some (princ,bindings) in let princ_vars = List.fold_right (fun a acc -> try Id.Set.add (destVar sigma a) acc with DestKO -> acc) args Id.Set.empty in let old_idl = List.fold_right Id.Set.add (Tacmach.pf_ids_of_hyps g) Id.Set.empty in let old_idl = Id.Set.diff old_idl princ_vars in let subst_and_reduce g = if with_clean then let idl = List.filter (fun id -> not (Id.Set.mem id old_idl)) (Tacmach.pf_ids_of_hyps g) in let flag = Genredexpr.Cbv {Redops.all_flags with Genredexpr.rDelta = false; } in Tacticals.tclTHEN (Tacticals.tclMAP (fun id -> Tacticals.tclTRY (Proofview.V82.of_tactic (Equality.subst_gen (do_rewrite_dependent ()) [id]))) idl ) (Proofview.V82.of_tactic (Tactics.reduce flag Locusops.allHypsAndConcl)) g else Tacticals.tclIDTAC g in Tacticals.tclTHEN (Proofview.V82.of_tactic (choose_dest_or_ind princ_infos (args_as_induction_constr,princ'))) subst_and_reduce g' in res let rec abstract_glob_constr c = function | [] -> c | Constrexpr.CLocalDef (x,b,t)::bl -> Constrexpr_ops.mkLetInC(x,b,t,abstract_glob_constr c bl) | Constrexpr.CLocalAssum (idl,k,t)::bl -> List.fold_right (fun x b -> Constrexpr_ops.mkLambdaC([x],k,t,b)) idl (abstract_glob_constr c bl) | Constrexpr.CLocalPattern _::bl -> assert false let interp_casted_constr_with_implicits env sigma impls c = Constrintern.intern_gen Pretyping.WithoutTypeConstraint env sigma ~impls c (* Construct a fixpoint as a Glob_term and not as a constr *) let build_newrecursive lnameargsardef = let env0 = Global.env() in let sigma = Evd.from_env env0 in let (rec_sign,rec_impls) = List.fold_left (fun (env,impls) { Vernacexpr.fname={CAst.v=recname}; binders; rtype } -> let arityc = Constrexpr_ops.mkCProdN binders rtype in let arity,ctx = Constrintern.interp_type env0 sigma arityc in let evd = Evd.from_env env0 in let evd, (_, (_, impls')) = Constrintern.interp_context_evars ~program_mode:false env evd binders in let impl = Constrintern.compute_internalization_data env0 evd Constrintern.Recursive arity impls' in let open Context.Named.Declaration in let r = Sorts.Relevant in (* TODO relevance *) (EConstr.push_named (LocalAssum (make_annot recname r,arity)) env, Id.Map.add recname impl impls)) (env0,Constrintern.empty_internalization_env) lnameargsardef in let recdef = (* Declare local notations *) let f { Vernacexpr.binders; body_def } = match body_def with | Some body_def -> let def = abstract_glob_constr body_def binders in interp_casted_constr_with_implicits rec_sign sigma rec_impls def | None -> user_err ~hdr:"Function" (str "Body of Function must be given") in States.with_state_protection (List.map f) lnameargsardef in recdef,rec_impls let error msg = user_err Pp.(str msg) (* Checks whether or not the mutual bloc is recursive *) let is_rec names = let names = List.fold_right Id.Set.add names Id.Set.empty in let check_id id names = Id.Set.mem id names in let rec lookup names gt = match DAst.get gt with | GVar(id) -> check_id id names | GRef _ | GEvar _ | GPatVar _ | GSort _ | GHole _ | GInt _ -> false | GCast(b,_) -> lookup names b | GRec _ -> error "GRec not handled" | GIf(b,_,lhs,rhs) -> (lookup names b) || (lookup names lhs) || (lookup names rhs) | GProd(na,_,t,b) | GLambda(na,_,t,b) -> lookup names t || lookup (Nameops.Name.fold_right Id.Set.remove na names) b | GLetIn(na,b,t,c) -> lookup names b || Option.cata (lookup names) true t || lookup (Nameops.Name.fold_right Id.Set.remove na names) c | GLetTuple(nal,_,t,b) -> lookup names t || lookup (List.fold_left (fun acc na -> Nameops.Name.fold_right Id.Set.remove na acc) names nal ) b | GApp(f,args) -> List.exists (lookup names) (f::args) | GCases(_,_,el,brl) -> List.exists (fun (e,_) -> lookup names e) el || List.exists (lookup_br names) brl and lookup_br names {CAst.v=(idl,_,rt)} = let new_names = List.fold_right Id.Set.remove idl names in lookup new_names rt in lookup names let rec local_binders_length = function (* Assume that no `{ ... } contexts occur *) | [] -> 0 | Constrexpr.CLocalDef _::bl -> 1 + local_binders_length bl | Constrexpr.CLocalAssum (idl,_,_)::bl -> List.length idl + local_binders_length bl | Constrexpr.CLocalPattern _::bl -> assert false let prepare_body { Vernacexpr.binders; rtype } rt = let n = local_binders_length binders in (* Pp.msgnl (str "nb lambda to chop : " ++ str (string_of_int n) ++ fnl () ++Printer.pr_glob_constr rt); *) let fun_args,rt' = chop_rlambda_n n rt in (fun_args,rt') (* [prove_fun_correct funs_constr graphs_constr schemes lemmas_types_infos i ] is the tactic used to prove correctness lemma. [funs_constr], [graphs_constr] [schemes] [lemmas_types_infos] are the mutually recursive functions (resp. graphs of the functions and principles and correctness lemma types) to prove correct. [i] is the indice of the function to prove correct The lemma to prove if suppose to have been generated by [generate_type] (in $\zeta$ normal form that is it looks like~: [\forall (x_1:t_1)\ldots(x_n:t_n), forall res, res = f x_1\ldots x_n in, \rightarrow graph\ x_1\ldots x_n\ res] The sketch of the proof is the following one~: \begin{enumerate} \item intros until $x_n$ \item $functional\ induction\ (f.(i)\ x_1\ldots x_n)$ using schemes.(i) \item for each generated branch intro [res] and [hres :res = f x_1\ldots x_n], rewrite [hres] and the apply the corresponding constructor of the corresponding graph inductive. \end{enumerate} *) let rec generate_fresh_id x avoid i = if i == 0 then [] else let id = Namegen.next_ident_away_in_goal x (Id.Set.of_list avoid) in id::(generate_fresh_id x (id::avoid) (pred i)) let make_eq () = try EConstr.of_constr (UnivGen.constr_of_monomorphic_global (Coqlib.lib_ref "core.eq.type")) with _ -> assert false let prove_fun_correct evd funs_constr graphs_constr schemes lemmas_types_infos i : Tacmach.tactic = let open Context.Rel.Declaration in let open Tacmach in let open Tactics in let open Tacticals in fun g -> (* first of all we recreate the lemmas types to be used as predicates of the induction principle that is~: \[fun (x_1:t_1)\ldots(x_n:t_n)=> fun fv => fun res => res = fv \rightarrow graph\ x_1\ldots x_n\ res\] *) (* we the get the definition of the graphs block *) let graph_ind,u = destInd evd graphs_constr.(i) in let kn = fst graph_ind in let mib,_ = Global.lookup_inductive graph_ind in (* and the principle to use in this lemma in $\zeta$ normal form *) let f_principle,princ_type = schemes.(i) in let princ_type = Reductionops.nf_zeta (Global.env ()) evd princ_type in let princ_infos = Tactics.compute_elim_sig evd princ_type in (* The number of args of the function is then easily computable *) let nb_fun_args = Termops.nb_prod (project g) (pf_concl g) - 2 in let args_names = generate_fresh_id (Id.of_string "x") [] nb_fun_args in let ids = args_names@(pf_ids_of_hyps g) in (* Since we cannot ensure that the functional principle is defined in the environment and due to the bug #1174, we will need to pose the principle using a name *) let principle_id = Namegen.next_ident_away_in_goal (Id.of_string "princ") (Id.Set.of_list ids) in let ids = principle_id :: ids in (* We get the branches of the principle *) let branches = List.rev princ_infos.Tactics.branches in (* and built the intro pattern for each of them *) let intro_pats = List.map (fun decl -> List.map (fun id -> CAst.make @@ IntroNaming (Namegen.IntroIdentifier id)) (generate_fresh_id (Id.of_string "y") ids (List.length (fst (decompose_prod_assum evd (RelDecl.get_type decl))))) ) branches in (* before building the full intro pattern for the principle *) let eq_ind = make_eq () in let eq_construct = mkConstructUi (destInd evd eq_ind, 1) in (* The next to referencies will be used to find out which constructor to apply in each branch *) let ind_number = ref 0 and min_constr_number = ref 0 in (* The tactic to prove the ith branch of the principle *) let prove_branche i g = (* We get the identifiers of this branch *) let pre_args = List.fold_right (fun {CAst.v=pat} acc -> match pat with | IntroNaming (Namegen.IntroIdentifier id) -> id::acc | _ -> anomaly (Pp.str "Not an identifier.") ) (List.nth intro_pats (pred i)) [] in (* and get the real args of the branch by unfolding the defined constant *) (* We can then recompute the arguments of the constructor. For each [hid] introduced by this branch, if [hid] has type $forall res, res=fv -> graph.(j)\ x_1\ x_n res$ the corresponding arguments of the constructor are [ fv (hid fv (refl_equal fv)) ]. If [hid] has another type the corresponding argument of the constructor is [hid] *) let constructor_args g = List.fold_right (fun hid acc -> let type_of_hid = pf_unsafe_type_of g (mkVar hid) in let sigma = project g in match EConstr.kind sigma type_of_hid with | Prod(_,_,t') -> begin match EConstr.kind sigma t' with | Prod(_,t'',t''') -> begin match EConstr.kind sigma t'',EConstr.kind sigma t''' with | App(eq,args), App(graph',_) when (EConstr.eq_constr sigma eq eq_ind) && Array.exists (EConstr.eq_constr_nounivs sigma graph') graphs_constr -> (args.(2)::(mkApp(mkVar hid,[|args.(2);(mkApp(eq_construct,[|args.(0);args.(2)|]))|])) ::acc) | _ -> mkVar hid :: acc end | _ -> mkVar hid :: acc end | _ -> mkVar hid :: acc ) pre_args [] in (* in fact we must also add the parameters to the constructor args *) let constructor_args g = let params_id = fst (List.chop princ_infos.Tactics.nparams args_names) in (List.map mkVar params_id)@((constructor_args g)) in (* We then get the constructor corresponding to this branch and modifies the references has needed i.e. if the constructor is the last one of the current inductive then add one the number of the inductive to take and add the number of constructor of the previous graph to the minimal constructor number *) let constructor = let constructor_num = i - !min_constr_number in let length = Array.length (mib.Declarations.mind_packets.(!ind_number).Declarations.mind_consnames) in if constructor_num <= length then begin (kn,!ind_number),constructor_num end else begin incr ind_number; min_constr_number := !min_constr_number + length ; (kn,!ind_number),1 end in (* we can then build the final proof term *) let app_constructor g = applist((mkConstructU(constructor,u)),constructor_args g) in (* an apply the tactic *) let res,hres = match generate_fresh_id (Id.of_string "z") (ids(* @this_branche_ids *)) 2 with | [res;hres] -> res,hres | _ -> assert false in (* observe (str "constructor := " ++ Printer.pr_lconstr_env (pf_env g) app_constructor); *) ( tclTHENLIST [ observe_tac("h_intro_patterns ") (let l = (List.nth intro_pats (pred i)) in match l with | [] -> tclIDTAC | _ -> Proofview.V82.of_tactic (intro_patterns false l)); (* unfolding of all the defined variables introduced by this branch *) (* observe_tac "unfolding" pre_tac; *) (* $zeta$ normalizing of the conclusion *) Proofview.V82.of_tactic (reduce (Genredexpr.Cbv { Redops.all_flags with Genredexpr.rDelta = false ; Genredexpr.rConst = [] } ) Locusops.onConcl); observe_tac ("toto ") tclIDTAC; (* introducing the result of the graph and the equality hypothesis *) observe_tac "introducing" (tclMAP (fun x -> Proofview.V82.of_tactic (Simple.intro x)) [res;hres]); (* replacing [res] with its value *) observe_tac "rewriting res value" (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar hres))); (* Conclusion *) observe_tac "exact" (fun g -> Proofview.V82.of_tactic (exact_check (app_constructor g)) g) ] ) g in (* end of branche proof *) let lemmas = Array.map (fun ((_,(ctxt,concl))) -> match ctxt with | [] | [_] | [_;_] -> anomaly (Pp.str "bad context.") | hres::res::decl::ctxt -> let res = EConstr.it_mkLambda_or_LetIn (EConstr.it_mkProd_or_LetIn concl [hres;res]) (LocalAssum (RelDecl.get_annot decl, RelDecl.get_type decl) :: ctxt) in res) lemmas_types_infos in let param_names = fst (List.chop princ_infos.nparams args_names) in let params = List.map mkVar param_names in let lemmas = Array.to_list (Array.map (fun c -> applist(c,params)) lemmas) in (* The bindings of the principle that is the params of the principle and the different lemma types *) let bindings = let params_bindings,avoid = List.fold_left2 (fun (bindings,avoid) decl p -> let id = Namegen.next_ident_away (Nameops.Name.get_id (RelDecl.get_name decl)) (Id.Set.of_list avoid) in p::bindings,id::avoid ) ([],pf_ids_of_hyps g) princ_infos.params (List.rev params) in let lemmas_bindings = List.rev (fst (List.fold_left2 (fun (bindings,avoid) decl p -> let id = Namegen.next_ident_away (Nameops.Name.get_id (RelDecl.get_name decl)) (Id.Set.of_list avoid) in (Reductionops.nf_zeta (pf_env g) (project g) p)::bindings,id::avoid) ([],avoid) princ_infos.predicates (lemmas))) in (params_bindings@lemmas_bindings) in tclTHENLIST [ observe_tac "principle" (Proofview.V82.of_tactic (assert_by (Name principle_id) princ_type (exact_check f_principle))); observe_tac "intro args_names" (tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) args_names); (* observe_tac "titi" (pose_proof (Name (Id.of_string "__")) (Reductionops.nf_beta Evd.empty ((mkApp (mkVar principle_id,Array.of_list bindings))))); *) observe_tac "idtac" tclIDTAC; tclTHEN_i (observe_tac "functional_induction" ( (fun gl -> let term = mkApp (mkVar principle_id,Array.of_list bindings) in let gl', _ty = pf_eapply (Typing.type_of ~refresh:true) gl term in Proofview.V82.of_tactic (apply term) gl') )) (fun i g -> observe_tac ("proving branche "^string_of_int i) (prove_branche i) g ) ] g (** [find_induction_principle f] searches and returns the [body] and the [type] of [f_rect] WARNING: while convertible, [type_of body] and [type] can be non equal *) let find_induction_principle evd f = let f_as_constant,u = match EConstr.kind !evd f with | Const c' -> c' | _ -> user_err Pp.(str "Must be used with a function") in let infos = find_Function_infos f_as_constant in match infos.rect_lemma with | None -> raise Not_found | Some rect_lemma -> let evd',rect_lemma = Evd.fresh_global (Global.env ()) !evd (GlobRef.ConstRef rect_lemma) in let evd',typ = Typing.type_of ~refresh:true (Global.env ()) evd' rect_lemma in evd:=evd'; rect_lemma,typ (* [generate_type g_to_f f graph i] build the completeness (resp. correctness) lemma type if [g_to_f = true] (resp. g_to_f = false) where [graph] is the graph of [f] and is the [i]th function in the block. [generate_type true f i] returns \[\forall (x_1:t_1)\ldots(x_n:t_n), let fv := f x_1\ldots x_n in, forall res, graph\ x_1\ldots x_n\ res \rightarrow res = fv \] decomposed as the context and the conclusion [generate_type false f i] returns \[\forall (x_1:t_1)\ldots(x_n:t_n), let fv := f x_1\ldots x_n in, forall res, res = fv \rightarrow graph\ x_1\ldots x_n\ res\] decomposed as the context and the conclusion *) let generate_type evd g_to_f f graph i = let open Context.Rel.Declaration in let open EConstr.Vars in (*i we deduce the number of arguments of the function and its returned type from the graph i*) let evd',graph = Evd.fresh_global (Global.env ()) !evd (GlobRef.IndRef (fst (destInd !evd graph))) in evd:=evd'; let sigma, graph_arity = Typing.type_of (Global.env ()) !evd graph in evd := sigma; let ctxt,_ = decompose_prod_assum !evd graph_arity in let fun_ctxt,res_type = match ctxt with | [] | [_] -> anomaly (Pp.str "Not a valid context.") | decl :: fun_ctxt -> fun_ctxt, RelDecl.get_type decl in let rec args_from_decl i accu = function | [] -> accu | LocalDef _ :: l -> args_from_decl (succ i) accu l | _ :: l -> let t = mkRel i in args_from_decl (succ i) (t :: accu) l in (*i We need to name the vars [res] and [fv] i*) let filter = fun decl -> match RelDecl.get_name decl with | Name id -> Some id | Anonymous -> None in let named_ctxt = Id.Set.of_list (List.map_filter filter fun_ctxt) in let res_id = Namegen.next_ident_away_in_goal (Id.of_string "_res") named_ctxt in let fv_id = Namegen.next_ident_away_in_goal (Id.of_string "fv") (Id.Set.add res_id named_ctxt) in (*i we can then type the argument to be applied to the function [f] i*) let args_as_rels = Array.of_list (args_from_decl 1 [] fun_ctxt) in (*i the hypothesis [res = fv] can then be computed We will need to lift it by one in order to use it as a conclusion i*) let make_eq = make_eq () in let res_eq_f_of_args = mkApp(make_eq ,[|lift 2 res_type;mkRel 1;mkRel 2|]) in (*i The hypothesis [graph\ x_1\ldots x_n\ res] can then be computed We will need to lift it by one in order to use it as a conclusion i*) let args_and_res_as_rels = Array.of_list (args_from_decl 3 [] fun_ctxt) in let args_and_res_as_rels = Array.append args_and_res_as_rels [|mkRel 1|] in let graph_applied = mkApp(graph, args_and_res_as_rels) in (*i The [pre_context] is the defined to be the context corresponding to \[\forall (x_1:t_1)\ldots(x_n:t_n), let fv := f x_1\ldots x_n in, forall res, \] i*) let pre_ctxt = LocalAssum (make_annot (Name res_id) Sorts.Relevant, lift 1 res_type) :: LocalDef (make_annot (Name fv_id) Sorts.Relevant, mkApp (f,args_as_rels), res_type) :: fun_ctxt in (*i and we can return the solution depending on which lemma type we are defining i*) if g_to_f then LocalAssum (make_annot Anonymous Sorts.Relevant,graph_applied)::pre_ctxt,(lift 1 res_eq_f_of_args),graph else LocalAssum (make_annot Anonymous Sorts.Relevant,res_eq_f_of_args)::pre_ctxt,(lift 1 graph_applied),graph (* [prove_fun_complete funs graphs schemes lemmas_types_infos i] is the tactic used to prove completeness lemma. [funcs], [graphs] [schemes] [lemmas_types_infos] are the mutually recursive functions (resp. definitions of the graphs of the functions, principles and correctness lemma types) to prove correct. [i] is the indice of the function to prove complete The lemma to prove if suppose to have been generated by [generate_type] (in $\zeta$ normal form that is it looks like~: [\forall (x_1:t_1)\ldots(x_n:t_n), forall res, graph\ x_1\ldots x_n\ res, \rightarrow res = f x_1\ldots x_n in] The sketch of the proof is the following one~: \begin{enumerate} \item intros until $H:graph\ x_1\ldots x_n\ res$ \item $elim\ H$ using schemes.(i) \item for each generated branch, intro the news hyptohesis, for each such hyptohesis [h], if [h] has type [x=?] with [x] a variable, then subst [x], if [h] has type [t=?] with [t] not a variable then rewrite [t] in the subterms, else if [h] is a match then destruct it, else do just introduce it, after all intros, the conclusion should be a reflexive equality. \end{enumerate} *) let thin ids gl = Proofview.V82.of_tactic (Tactics.clear ids) gl (* [intros_with_rewrite] do the intros in each branch and treat each new hypothesis (unfolding, substituting, destructing cases \ldots) *) let tauto = let open Ltac_plugin in let dp = List.map Id.of_string ["Tauto" ; "Init"; "Coq"] in let mp = ModPath.MPfile (DirPath.make dp) in let kn = KerName.make mp (Label.make "tauto") in Proofview.tclBIND (Proofview.tclUNIT ()) begin fun () -> let body = Tacenv.interp_ltac kn in Tacinterp.eval_tactic body end (* [generalize_dependent_of x hyp g] generalize every hypothesis which depends of [x] but [hyp] *) let generalize_dependent_of x hyp g = let open Context.Named.Declaration in let open Tacmach in let open Tacticals in tclMAP (function | LocalAssum ({binder_name=id},t) when not (Id.equal id hyp) && (Termops.occur_var (pf_env g) (project g) x t) -> tclTHEN (Proofview.V82.of_tactic (Tactics.generalize [mkVar id])) (thin [id]) | _ -> tclIDTAC ) (pf_hyps g) g let rec intros_with_rewrite g = observe_tac "intros_with_rewrite" intros_with_rewrite_aux g and intros_with_rewrite_aux : Tacmach.tactic = let open Tacmach in let open Tactics in let open Tacticals in fun g -> let eq_ind = make_eq () in let sigma = project g in match EConstr.kind sigma (pf_concl g) with | Prod(_,t,t') -> begin match EConstr.kind sigma t with | App(eq,args) when (EConstr.eq_constr sigma eq eq_ind) -> if Reductionops.is_conv (pf_env g) (project g) args.(1) args.(2) then let id = pf_get_new_id (Id.of_string "y") g in tclTHENLIST [ Proofview.V82.of_tactic (Simple.intro id); thin [id]; intros_with_rewrite ] g else if isVar sigma args.(1) && (Environ.evaluable_named (destVar sigma args.(1)) (pf_env g)) then tclTHENLIST[ Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, Names.EvalVarRef (destVar sigma args.(1)))]); tclMAP (fun id -> tclTRY(Proofview.V82.of_tactic (unfold_in_hyp [(Locus.AllOccurrences, Names.EvalVarRef (destVar sigma args.(1)))] ((destVar sigma args.(1)),Locus.InHyp) ))) (pf_ids_of_hyps g); intros_with_rewrite ] g else if isVar sigma args.(2) && (Environ.evaluable_named (destVar sigma args.(2)) (pf_env g)) then tclTHENLIST[ Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, Names.EvalVarRef (destVar sigma args.(2)))]); tclMAP (fun id -> tclTRY(Proofview.V82.of_tactic (unfold_in_hyp [(Locus.AllOccurrences, Names.EvalVarRef (destVar sigma args.(2)))] ((destVar sigma args.(2)),Locus.InHyp) ))) (pf_ids_of_hyps g); intros_with_rewrite ] g else if isVar sigma args.(1) then let id = pf_get_new_id (Id.of_string "y") g in tclTHENLIST [ Proofview.V82.of_tactic (Simple.intro id); generalize_dependent_of (destVar sigma args.(1)) id; tclTRY (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar id))); intros_with_rewrite ] g else if isVar sigma args.(2) then let id = pf_get_new_id (Id.of_string "y") g in tclTHENLIST [ Proofview.V82.of_tactic (Simple.intro id); generalize_dependent_of (destVar sigma args.(2)) id; tclTRY (Proofview.V82.of_tactic (Equality.rewriteRL (mkVar id))); intros_with_rewrite ] g else begin let id = pf_get_new_id (Id.of_string "y") g in tclTHENLIST[ Proofview.V82.of_tactic (Simple.intro id); tclTRY (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar id))); intros_with_rewrite ] g end | Ind _ when EConstr.eq_constr sigma t (EConstr.of_constr (UnivGen.constr_of_monomorphic_global @@ Coqlib.lib_ref "core.False.type")) -> Proofview.V82.of_tactic tauto g | Case(_,_,v,_) -> tclTHENLIST[ Proofview.V82.of_tactic (simplest_case v); intros_with_rewrite ] g | LetIn _ -> tclTHENLIST[ Proofview.V82.of_tactic (reduce (Genredexpr.Cbv {Redops.all_flags with Genredexpr.rDelta = false; }) Locusops.onConcl) ; intros_with_rewrite ] g | _ -> let id = pf_get_new_id (Id.of_string "y") g in tclTHENLIST [ Proofview.V82.of_tactic (Simple.intro id);intros_with_rewrite] g end | LetIn _ -> tclTHENLIST[ Proofview.V82.of_tactic (reduce (Genredexpr.Cbv {Redops.all_flags with Genredexpr.rDelta = false; }) Locusops.onConcl) ; intros_with_rewrite ] g | _ -> tclIDTAC g let rec reflexivity_with_destruct_cases g = let open Tacmach in let open Tactics in let open Tacticals in let destruct_case () = try match EConstr.kind (project g) (snd (destApp (project g) (pf_concl g))).(2) with | Case(_,_,v,_) -> tclTHENLIST[ Proofview.V82.of_tactic (simplest_case v); Proofview.V82.of_tactic intros; observe_tac "reflexivity_with_destruct_cases" reflexivity_with_destruct_cases ] | _ -> Proofview.V82.of_tactic reflexivity with e when CErrors.noncritical e -> Proofview.V82.of_tactic reflexivity in let eq_ind = make_eq () in let my_inj_flags = Some { Equality.keep_proof_equalities = false; injection_in_context = false; (* for compatibility, necessary *) injection_pattern_l2r_order = false; (* probably does not matter; except maybe with dependent hyps *) } in let discr_inject = Tacticals.onAllHypsAndConcl ( fun sc g -> match sc with None -> tclIDTAC g | Some id -> match EConstr.kind (project g) (pf_unsafe_type_of g (mkVar id)) with | App(eq,[|_;t1;t2|]) when EConstr.eq_constr (project g) eq eq_ind -> if Equality.discriminable (pf_env g) (project g) t1 t2 then Proofview.V82.of_tactic (Equality.discrHyp id) g else if Equality.injectable (pf_env g) (project g) ~keep_proofs:None t1 t2 then tclTHENLIST [Proofview.V82.of_tactic (Equality.injHyp my_inj_flags None id);thin [id];intros_with_rewrite] g else tclIDTAC g | _ -> tclIDTAC g ) in (tclFIRST [ observe_tac "reflexivity_with_destruct_cases : reflexivity" (Proofview.V82.of_tactic reflexivity); observe_tac "reflexivity_with_destruct_cases : destruct_case" ((destruct_case ())); (* We reach this point ONLY if the same value is matched (at least) two times along binding path. In this case, either we have a discriminable hypothesis and we are done, either at least an injectable one and we do the injection before continuing *) observe_tac "reflexivity_with_destruct_cases : others" (tclTHEN (tclPROGRESS discr_inject ) reflexivity_with_destruct_cases) ]) g let prove_fun_complete funcs graphs schemes lemmas_types_infos i : Tacmach.tactic = let open Tacmach in let open Tactics in let open Tacticals in fun g -> (* We compute the types of the different mutually recursive lemmas in $\zeta$ normal form *) let lemmas = Array.map (fun (_,(ctxt,concl)) -> Reductionops.nf_zeta (pf_env g) (project g) (EConstr.it_mkLambda_or_LetIn concl ctxt)) lemmas_types_infos in (* We get the constant and the principle corresponding to this lemma *) let f = funcs.(i) in let graph_principle = Reductionops.nf_zeta (pf_env g) (project g) (EConstr.of_constr schemes.(i)) in let princ_type = pf_unsafe_type_of g graph_principle in let princ_infos = Tactics.compute_elim_sig (project g) princ_type in (* Then we get the number of argument of the function and compute a fresh name for each of them *) let nb_fun_args = Termops.nb_prod (project g) (pf_concl g) - 2 in let args_names = generate_fresh_id (Id.of_string "x") [] nb_fun_args in let ids = args_names@(pf_ids_of_hyps g) in (* and fresh names for res H and the principle (cf bug bug #1174) *) let res,hres,graph_principle_id = match generate_fresh_id (Id.of_string "z") ids 3 with | [res;hres;graph_principle_id] -> res,hres,graph_principle_id | _ -> assert false in let ids = res::hres::graph_principle_id::ids in (* we also compute fresh names for each hyptohesis of each branch of the principle *) let branches = List.rev princ_infos.branches in let intro_pats = List.map (fun decl -> List.map (fun id -> id) (generate_fresh_id (Id.of_string "y") ids (Termops.nb_prod (project g) (RelDecl.get_type decl))) ) branches in (* We will need to change the function by its body using [f_equation] if it is recursive (that is the graph is infinite or unfold if the graph is finite *) let rewrite_tac j ids : Tacmach.tactic = let graph_def = graphs.(j) in let infos = try find_Function_infos (fst (destConst (project g) funcs.(j))) with Not_found -> user_err Pp.(str "No graph found") in if infos.is_general || Rtree.is_infinite Declareops.eq_recarg graph_def.mind_recargs then let eq_lemma = try Option.get (infos).equation_lemma with Option.IsNone -> anomaly (Pp.str "Cannot find equation lemma.") in tclTHENLIST[ tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) ids; Proofview.V82.of_tactic (Equality.rewriteLR (mkConst eq_lemma)); (* Don't forget to $\zeta$ normlize the term since the principles have been $\zeta$-normalized *) Proofview.V82.of_tactic (reduce (Genredexpr.Cbv {Redops.all_flags with Genredexpr.rDelta = false; }) Locusops.onConcl) ; Proofview.V82.of_tactic (generalize (List.map mkVar ids)); thin ids ] else Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, Names.EvalConstRef (fst (destConst (project g) f)))]) in (* The proof of each branche itself *) let ind_number = ref 0 in let min_constr_number = ref 0 in let prove_branche i g = (* we fist compute the inductive corresponding to the branch *) let this_ind_number = let constructor_num = i - !min_constr_number in let length = Array.length (graphs.(!ind_number).Declarations.mind_consnames) in if constructor_num <= length then !ind_number else begin incr ind_number; min_constr_number := !min_constr_number + length; !ind_number end in let this_branche_ids = List.nth intro_pats (pred i) in tclTHENLIST[ (* we expand the definition of the function *) observe_tac "rewrite_tac" (rewrite_tac this_ind_number this_branche_ids); (* introduce hypothesis with some rewrite *) observe_tac "intros_with_rewrite (all)" intros_with_rewrite; (* The proof is (almost) complete *) observe_tac "reflexivity" (reflexivity_with_destruct_cases) ] g in let params_names = fst (List.chop princ_infos.nparams args_names) in let open EConstr in let params = List.map mkVar params_names in tclTHENLIST [ tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) (args_names@[res;hres]); observe_tac "h_generalize" (Proofview.V82.of_tactic (generalize [mkApp(applist(graph_principle,params),Array.map (fun c -> applist(c,params)) lemmas)])); Proofview.V82.of_tactic (Simple.intro graph_principle_id); observe_tac "" (tclTHEN_i (observe_tac "elim" (Proofview.V82.of_tactic (elim false None (mkVar hres,NoBindings) (Some (mkVar graph_principle_id,NoBindings))))) (fun i g -> observe_tac "prove_branche" (prove_branche i) g )) ] g (* [derive_correctness make_scheme funs graphs] create correctness and completeness lemmas for each function in [funs] w.r.t. [graphs] [make_scheme] is Functional_principle_types.make_scheme (dependency pb) and *) let derive_correctness (funs: pconstant list) (graphs:inductive list) = assert (funs <> []); assert (graphs <> []); let funs = Array.of_list funs and graphs = Array.of_list graphs in let map (c, u) = mkConstU (c, EInstance.make u) in let funs_constr = Array.map map funs in (* XXX STATE Why do we need this... why is the toplevel protection not enough *) funind_purify (fun () -> let env = Global.env () in let evd = ref (Evd.from_env env) in let graphs_constr = Array.map mkInd graphs in let lemmas_types_infos = Util.Array.map2_i (fun i f_constr graph -> (* let const_of_f,u = destConst f_constr in *) let (type_of_lemma_ctxt,type_of_lemma_concl,graph) = generate_type evd false f_constr graph i in let type_info = (type_of_lemma_ctxt,type_of_lemma_concl) in graphs_constr.(i) <- graph; let type_of_lemma = EConstr.it_mkProd_or_LetIn type_of_lemma_concl type_of_lemma_ctxt in let sigma, _ = Typing.type_of (Global.env ()) !evd type_of_lemma in evd := sigma; let type_of_lemma = Reductionops.nf_zeta (Global.env ()) !evd type_of_lemma in observe (str "type_of_lemma := " ++ Printer.pr_leconstr_env (Global.env ()) !evd type_of_lemma); type_of_lemma,type_info ) funs_constr graphs_constr in let schemes = (* The functional induction schemes are computed and not saved if there is more that one function if the block contains only one function we can safely reuse [f_rect] *) try if not (Int.equal (Array.length funs_constr) 1) then raise Not_found; [| find_induction_principle evd funs_constr.(0) |] with Not_found -> ( Array.of_list (List.map (fun entry -> (EConstr.of_constr (fst (fst(Future.force entry.Proof_global.proof_entry_body))), EConstr.of_constr (Option.get entry.Proof_global.proof_entry_type )) ) (Functional_principles_types.make_scheme evd (Array.map_to_list (fun const -> const,Sorts.InType) funs)) ) ) in let proving_tac = prove_fun_correct !evd funs_constr graphs_constr schemes lemmas_types_infos in Array.iteri (fun i f_as_constant -> let f_id = Label.to_id (Constant.label (fst f_as_constant)) in (*i The next call to mk_correct_id is valid since we are constructing the lemma Ensures by: obvious i*) let lem_id = mk_correct_id f_id in let (typ,_) = lemmas_types_infos.(i) in let info = Lemmas.Info.make ~scope:(DeclareDef.Global Declare.ImportDefaultBehavior) ~kind:(Decls.(IsProof Theorem)) () in let lemma = Lemmas.start_lemma ~name:lem_id ~poly:false ~info !evd typ in let lemma = fst @@ Lemmas.by (Proofview.V82.tactic (proving_tac i)) lemma in let () = Lemmas.save_lemma_proved ~lemma ~opaque:Proof_global.Transparent ~idopt:None in let finfo = find_Function_infos (fst f_as_constant) in (* let lem_cst = fst (destConst (Constrintern.global_reference lem_id)) in *) let _,lem_cst_constr = Evd.fresh_global (Global.env ()) !evd (Constrintern.locate_reference (Libnames.qualid_of_ident lem_id)) in let (lem_cst,_) = destConst !evd lem_cst_constr in update_Function {finfo with correctness_lemma = Some lem_cst}; ) funs; let lemmas_types_infos = Util.Array.map2_i (fun i f_constr graph -> let (type_of_lemma_ctxt,type_of_lemma_concl,graph) = generate_type evd true f_constr graph i in let type_info = (type_of_lemma_ctxt,type_of_lemma_concl) in graphs_constr.(i) <- graph; let type_of_lemma = EConstr.it_mkProd_or_LetIn type_of_lemma_concl type_of_lemma_ctxt in let type_of_lemma = Reductionops.nf_zeta env !evd type_of_lemma in observe (str "type_of_lemma := " ++ Printer.pr_leconstr_env env !evd type_of_lemma); type_of_lemma,type_info ) funs_constr graphs_constr in let (kn,_) as graph_ind,u = (destInd !evd graphs_constr.(0)) in let mib,mip = Global.lookup_inductive graph_ind in let sigma, scheme = (Indrec.build_mutual_induction_scheme (Global.env ()) !evd (Array.to_list (Array.mapi (fun i _ -> ((kn,i), EInstance.kind !evd u),true,InType) mib.Declarations.mind_packets ) ) ) in let schemes = Array.of_list scheme in let proving_tac = prove_fun_complete funs_constr mib.Declarations.mind_packets schemes lemmas_types_infos in Array.iteri (fun i f_as_constant -> let f_id = Label.to_id (Constant.label (fst f_as_constant)) in (*i The next call to mk_complete_id is valid since we are constructing the lemma Ensures by: obvious i*) let lem_id = mk_complete_id f_id in let info = Lemmas.Info.make ~scope:(DeclareDef.Global Declare.ImportDefaultBehavior) ~kind:Decls.(IsProof Theorem) () in let lemma = Lemmas.start_lemma ~name:lem_id ~poly:false ~info sigma (fst lemmas_types_infos.(i)) in let lemma = fst (Lemmas.by (Proofview.V82.tactic (observe_tac ("prove completeness ("^(Id.to_string f_id)^")") (proving_tac i))) lemma) in let () = Lemmas.save_lemma_proved ~lemma ~opaque:Proof_global.Transparent ~idopt:None in let finfo = find_Function_infos (fst f_as_constant) in let _,lem_cst_constr = Evd.fresh_global (Global.env ()) !evd (Constrintern.locate_reference (Libnames.qualid_of_ident lem_id)) in let (lem_cst,_) = destConst !evd lem_cst_constr in update_Function {finfo with completeness_lemma = Some lem_cst} ) funs) () let warn_funind_cannot_build_inversion = CWarnings.create ~name:"funind-cannot-build-inversion" ~category:"funind" (fun e' -> strbrk "Cannot build inversion information" ++ if do_observe () then (fnl() ++ CErrors.print e') else mt ()) let derive_inversion fix_names = try let evd' = Evd.from_env (Global.env ()) in (* we first transform the fix_names identifier into their corresponding constant *) let evd',fix_names_as_constant = List.fold_right (fun id (evd,l) -> let evd,c = Evd.fresh_global (Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident id)) in let (cst, u) = destConst evd c in evd, (cst, EInstance.kind evd u) :: l ) fix_names (evd',[]) 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 (fst c))) fix_names_as_constant ; try let evd', lind = List.fold_right (fun id (evd,l) -> let evd,id = Evd.fresh_global (Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident (mk_rel_id id))) in evd,(fst (destInd evd id))::l ) fix_names (evd',[]) in derive_correctness fix_names_as_constant lind; with e when CErrors.noncritical e -> warn_funind_cannot_build_inversion e with e when CErrors.noncritical e -> warn_funind_cannot_build_inversion e let warn_cannot_define_graph = CWarnings.create ~name:"funind-cannot-define-graph" ~category:"funind" (fun (names,error) -> strbrk "Cannot define graph(s) for " ++ h 1 names ++ error) let warn_cannot_define_principle = CWarnings.create ~name:"funind-cannot-define-principle" ~category:"funind" (fun (names,error) -> strbrk "Cannot define induction principle(s) for "++ h 1 names ++ error) let warning_error names e = let e_explain e = match e with | ToShow e -> spc () ++ CErrors.print e | _ -> if do_observe () then (spc () ++ CErrors.print e) else mt () in match e with | Building_graph e -> let names = prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names in warn_cannot_define_graph (names,e_explain e) | Defining_principle e -> let names = prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names in warn_cannot_define_principle (names,e_explain e) | _ -> raise e let error_error names e = let e_explain e = match e with | ToShow e -> spc () ++ CErrors.print e | _ -> if do_observe () then (spc () ++ CErrors.print e) else mt () in match e with | Building_graph e -> user_err (str "Cannot define graph(s) for " ++ h 1 (prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names) ++ e_explain e) | _ -> raise e let generate_principle (evd:Evd.evar_map ref) pconstants on_error is_general do_built (fix_rec_l : Vernacexpr.fixpoint_expr list) recdefs interactive_proof (continue_proof : int -> Names.Constant.t array -> EConstr.constr array -> int -> Tacmach.tactic) : unit = let names = List.map (function { Vernacexpr.fname = {CAst.v=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 { Vernacexpr.rtype } -> rtype) fix_rec_l in try (* We then register the Inductive graphs of the functions *) Glob_term_to_relation.build_inductive !evd pconstants 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 = qualid_of_ident @@ mk_rel_id (List.nth names 0) in let ind_kn = fst (locate_with_msg (pr_qualid f_R_mut++str ": Not an inductive type!") locate_ind f_R_mut) in let fname_kn { Vernacexpr.fname } = let f_ref = qualid_of_ident ?loc:fname.CAst.loc fname.CAst.v in locate_with_msg (pr_qualid 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 env = Global.env () in let princ = Indrec.lookup_eliminator env (ind_kn,i) (InProp) in let evd = ref (Evd.from_env env) in let evd',uprinc = Evd.fresh_global env !evd princ in let _ = evd := evd' in let sigma, princ_type = Typing.type_of ~refresh:true env !evd uprinc in evd := sigma; let princ_type = EConstr.Unsafe.to_constr princ_type in Functional_principles_types.generate_functional_principle evd interactive_proof princ_type None None (Array.of_list pconstants) (* 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 when CErrors.noncritical e -> on_error names e let register_struct is_rec (fixpoint_exprl: Vernacexpr.fixpoint_expr list) = match fixpoint_exprl with | [ { Vernacexpr.fname; univs; binders; rtype; body_def } ] when not is_rec -> let body = match body_def with | Some body -> body | None -> user_err ~hdr:"Function" (str "Body of Function must be given") in ComDefinition.do_definition ~program_mode:false ~name:fname.CAst.v ~poly:false ~scope:(DeclareDef.Global Declare.ImportDefaultBehavior) ~kind:Decls.Definition univs binders None body (Some rtype); let evd,rev_pconstants = List.fold_left (fun (evd,l) { Vernacexpr.fname } -> let evd,c = Evd.fresh_global (Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident fname.CAst.v)) in let (cst, u) = destConst evd c in let u = EInstance.kind evd u in evd,((cst, u) :: l) ) (Evd.from_env (Global.env ()),[]) fixpoint_exprl in None, evd,List.rev rev_pconstants | _ -> ComFixpoint.do_fixpoint ~scope:(DeclareDef.Global Declare.ImportDefaultBehavior) ~poly:false fixpoint_exprl; let evd,rev_pconstants = List.fold_left (fun (evd,l) { Vernacexpr.fname } -> let evd,c = Evd.fresh_global (Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident fname.CAst.v)) in let (cst, u) = destConst evd c in let u = EInstance.kind evd u in evd,((cst, u) :: l) ) (Evd.from_env (Global.env ()),[]) fixpoint_exprl in None,evd,List.rev rev_pconstants 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.t array) (_:EConstr.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 interactive_proof ?(is_mes=false) fname rec_impls wf_rel_expr wf_arg using_lemmas args ret_type body pre_hook = let type_of_f = Constrexpr_ops.mkCProdN args ret_type in let rec_arg_num = let names = List.map CAst.(with_val (fun x -> x)) (Constrexpr_ops.names_of_local_assums args) in List.index Name.equal (Name wf_arg) names in let unbounded_eq = let f_app_args = CAst.make @@ Constrexpr.CAppExpl( (None,qualid_of_ident fname.CAst.v,None) , (List.map (function | {CAst.v=Anonymous} -> assert false | {CAst.v=Name e} -> (Constrexpr_ops.mkIdentC e) ) (Constrexpr_ops.names_of_local_assums args) ) ) in CAst.make @@ Constrexpr.CApp ((None,Constrexpr_ops.mkRefC (qualid_of_string "Logic.eq")), [(f_app_args,None);(body,None)]) in let eq = Constrexpr_ops.mkCProdN args unbounded_eq in let hook ((f_ref,_) as fconst) tcc_lemma_ref (functional_ref,_) (eq_ref,_) rec_arg_num rec_arg_type nb_args relation = try pre_hook [fconst] (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.CAst.v] with e when CErrors.noncritical e -> (* No proof done *) () in Recdef.recursive_definition ~interactive_proof ~is_mes fname.CAst.v rec_impls type_of_f wf_rel_expr rec_arg_num eq hook using_lemmas let register_mes interactive_proof fname rec_impls wf_mes_expr wf_rel_expr_opt wf_arg using_lemmas args ret_type body = let wf_arg_type,wf_arg = match wf_arg with | None -> begin match args with | [Constrexpr.CLocalAssum ([{CAst.v=Name x}],k,t)] -> t,x | _ -> error "Recursive argument must be specified" end | Some wf_args -> try match List.find (function | Constrexpr.CLocalAssum(l,k,t) -> List.exists (function {CAst.v=Name id} -> Id.equal id wf_args | _ -> false) l | _ -> false ) args with | Constrexpr.CLocalAssum(_,k,t) -> t,wf_args | _ -> assert false with Not_found -> assert false in let wf_rel_from_mes,is_mes = match wf_rel_expr_opt with | None -> let ltof = let make_dir l = DirPath.make (List.rev_map Id.of_string l) in Libnames.qualid_of_path (Libnames.make_path (make_dir ["Arith";"Wf_nat"]) (Id.of_string "ltof")) in let fun_from_mes = let applied_mes = Constrexpr_ops.mkAppC(wf_mes_expr,[Constrexpr_ops.mkIdentC wf_arg]) in Constrexpr_ops.mkLambdaC ([CAst.make @@ Name wf_arg],Constrexpr_ops.default_binder_kind,wf_arg_type,applied_mes) in let wf_rel_from_mes = Constrexpr_ops.mkAppC(Constrexpr_ops.mkRefC ltof,[wf_arg_type;fun_from_mes]) in wf_rel_from_mes,true | Some wf_rel_expr -> let wf_rel_with_mes = let a = Names.Id.of_string "___a" in let b = Names.Id.of_string "___b" in Constrexpr_ops.mkLambdaC( [CAst.make @@ Name a; CAst.make @@ Name b], Constrexpr.Default Explicit, wf_arg_type, Constrexpr_ops.mkAppC(wf_rel_expr, [ Constrexpr_ops.mkAppC(wf_mes_expr,[Constrexpr_ops.mkIdentC a]); Constrexpr_ops.mkAppC(wf_mes_expr,[Constrexpr_ops.mkIdentC b]) ]) ) in wf_rel_with_mes,false in register_wf interactive_proof ~is_mes:is_mes fname rec_impls wf_rel_from_mes wf_arg using_lemmas args ret_type body let map_option f = function | None -> None | Some v -> Some (f v) open Constrexpr let rec rebuild_bl aux bl typ = match bl,typ with | [], _ -> List.rev aux,typ | (CLocalAssum(nal,bk,_))::bl',typ -> rebuild_nal aux bk bl' nal typ | (CLocalDef(na,_,_))::bl',{ CAst.v = CLetIn(_,nat,ty,typ') } -> rebuild_bl (Constrexpr.CLocalDef(na,nat,ty)::aux) bl' typ' | _ -> assert false and rebuild_nal aux bk bl' nal typ = match nal,typ with | _,{ CAst.v = CProdN([],typ) } -> rebuild_nal aux bk bl' nal typ | [], _ -> rebuild_bl aux bl' typ | na::nal,{ CAst.v = CProdN(CLocalAssum(na'::nal',bk',nal't)::rest,typ') } -> if Name.equal (na.CAst.v) (na'.CAst.v) || Name.is_anonymous (na'.CAst.v) then let assum = CLocalAssum([na],bk,nal't) in let new_rest = if nal' = [] then rest else (CLocalAssum(nal',bk',nal't)::rest) in rebuild_nal (assum::aux) bk bl' nal (CAst.make @@ CProdN(new_rest,typ')) else let assum = CLocalAssum([na'],bk,nal't) in let new_rest = if nal' = [] then rest else (CLocalAssum(nal',bk',nal't)::rest) in rebuild_nal (assum::aux) bk bl' (na::nal) (CAst.make @@ CProdN(new_rest,typ')) | _ -> assert false let rebuild_bl aux bl typ = rebuild_bl aux bl typ let recompute_binder_list fixpoint_exprl = let fixl = List.map (fun fix -> Vernacexpr.{ fix with rec_order = ComFixpoint.adjust_rec_order ~structonly:false fix.binders fix.rec_order }) fixpoint_exprl in let ((_,_,_,typel),_,ctx,_) = ComFixpoint.interp_fixpoint ~cofix:false fixl in let constr_expr_typel = with_full_print (List.map (fun c -> Constrextern.extern_constr false (Global.env ()) (Evd.from_ctx ctx) (EConstr.of_constr c))) typel in let fixpoint_exprl_with_new_bl = List.map2 (fun ({ Vernacexpr.binders } as fp) fix_typ -> let binders, rtype = rebuild_bl [] binders fix_typ in { fp with Vernacexpr.binders; rtype } ) fixpoint_exprl constr_expr_typel in fixpoint_exprl_with_new_bl let do_generate_principle_aux pconstants on_error register_built interactive_proof (fixpoint_exprl : Vernacexpr.fixpoint_expr list) : Lemmas.t option = List.iter (fun { Vernacexpr.notations } -> if not (List.is_empty notations) then error "Function does not support notations for now") fixpoint_exprl; let lemma, _is_struct = match fixpoint_exprl with | [{ Vernacexpr.rec_order = Some {CAst.v = Constrexpr.CWfRec (wf_x,wf_rel)} } as fixpoint_expr] -> let { Vernacexpr.fname; univs; binders; rtype; body_def } as fixpoint_expr = match recompute_binder_list [fixpoint_expr] with | [e] -> e | _ -> assert false in let fixpoint_exprl = [fixpoint_expr] in let body = match body_def with | Some body -> body | None -> user_err ~hdr:"Function" (str "Body of Function must be given") in let recdefs,rec_impls = build_newrecursive fixpoint_exprl in let using_lemmas = [] in let pre_hook pconstants = generate_principle (ref (Evd.from_env (Global.env ()))) pconstants on_error true register_built fixpoint_exprl recdefs true in if register_built then register_wf interactive_proof fname rec_impls wf_rel wf_x.CAst.v using_lemmas binders rtype body pre_hook, false else None, false |[{ Vernacexpr.rec_order=Some {CAst.v = Constrexpr.CMeasureRec(wf_x,wf_mes,wf_rel_opt)} } as fixpoint_expr] -> let { Vernacexpr.fname; univs; binders; rtype; body_def} as fixpoint_expr = match recompute_binder_list [fixpoint_expr] with | [e] -> e | _ -> assert false in let fixpoint_exprl = [fixpoint_expr] in let recdefs,rec_impls = build_newrecursive fixpoint_exprl in let using_lemmas = [] in let body = match body_def with | Some body -> body | None -> user_err ~hdr:"Function" (str "Body of Function must be given") in let pre_hook pconstants = generate_principle (ref (Evd.from_env (Global.env ()))) pconstants on_error true register_built fixpoint_exprl recdefs true in if register_built then register_mes interactive_proof fname rec_impls wf_mes wf_rel_opt (map_option (fun x -> x.CAst.v) wf_x) using_lemmas binders rtype body pre_hook, true else None, true | _ -> List.iter (function { Vernacexpr.rec_order } -> match rec_order with | Some { CAst.v = (Constrexpr.CMeasureRec _ | Constrexpr.CWfRec _) } -> error ("Cannot use mutual definition with well-founded recursion or measure") | _ -> () ) fixpoint_exprl; let fixpoint_exprl = recompute_binder_list fixpoint_exprl in let fix_names = List.map (function { Vernacexpr.fname } -> fname.CAst.v) fixpoint_exprl in (* ok all the expressions are structural *) let recdefs,rec_impls = build_newrecursive fixpoint_exprl in let is_rec = List.exists (is_rec fix_names) recdefs in let lemma,evd,pconstants = if register_built then register_struct is_rec fixpoint_exprl else None, Evd.from_env (Global.env ()), pconstants in let evd = ref evd in generate_principle (ref !evd) pconstants on_error false register_built fixpoint_exprl recdefs interactive_proof (Functional_principles_proofs.prove_princ_for_struct evd interactive_proof); if register_built then begin derive_inversion fix_names; end; lemma, true in lemma let rec add_args id new_args = CAst.map (function | CRef (qid,_) as b -> if qualid_is_ident qid && Id.equal (qualid_basename qid) id then CAppExpl((None,qid,None),new_args) else b | CFix _ | CCoFix _ -> anomaly ~label:"add_args " (Pp.str "todo.") | CProdN(nal,b1) -> CProdN(List.map (function CLocalAssum (nal,k,b2) -> CLocalAssum (nal,k,add_args id new_args b2) | CLocalDef (na,b1,t) -> CLocalDef (na,add_args id new_args b1,Option.map (add_args id new_args) t) | CLocalPattern _ -> user_err (Pp.str "pattern with quote not allowed here.")) nal, add_args id new_args b1) | CLambdaN(nal,b1) -> CLambdaN(List.map (function CLocalAssum (nal,k,b2) -> CLocalAssum (nal,k,add_args id new_args b2) | CLocalDef (na,b1,t) -> CLocalDef (na,add_args id new_args b1,Option.map (add_args id new_args) t) | CLocalPattern _ -> user_err (Pp.str "pattern with quote not allowed here.")) nal, add_args id new_args b1) | CLetIn(na,b1,t,b2) -> CLetIn(na,add_args id new_args b1,Option.map (add_args id new_args) t,add_args id new_args b2) | CAppExpl((pf,qid,us),exprl) -> if qualid_is_ident qid && Id.equal (qualid_basename qid) id then CAppExpl((pf,qid,us),new_args@(List.map (add_args id new_args) exprl)) else CAppExpl((pf,qid,us),List.map (add_args id new_args) exprl) | CApp((pf,b),bl) -> CApp((pf,add_args id new_args b), List.map (fun (e,o) -> add_args id new_args e,o) bl) | CCases(sty,b_option,cel,cal) -> CCases(sty,Option.map (add_args id new_args) b_option, List.map (fun (b,na,b_option) -> add_args id new_args b, na, b_option) cel, List.map CAst.(map (fun (cpl,e) -> (cpl,add_args id new_args e))) cal ) | CLetTuple(nal,(na,b_option),b1,b2) -> CLetTuple(nal,(na,Option.map (add_args id new_args) b_option), add_args id new_args b1, add_args id new_args b2 ) | CIf(b1,(na,b_option),b2,b3) -> CIf(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 _ | CPatVar _ | CEvar _ | CPrim _ | CSort _ as b -> b | CCast(b1,b2) -> CCast(add_args id new_args b1, Glob_ops.map_cast_type (add_args id new_args) b2) | CRecord pars -> CRecord (List.map (fun (e,o) -> e, add_args id new_args o) pars) | CNotation _ -> anomaly ~label:"add_args " (Pp.str "CNotation.") | CGeneralization _ -> anomaly ~label:"add_args " (Pp.str "CGeneralization.") | CDelimiters _ -> anomaly ~label:"add_args " (Pp.str "CDelimiters.") ) exception Stop of Constrexpr.constr_expr (* [chop_n_arrow n t] chops the [n] first arrows in [t] Acts on Constrexpr.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.CAst.v with | Constrexpr.CProdN(nal_ta',t') -> (* If we have a forall, two results 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 | CLocalAssum(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' = CAst.make @@ Constrexpr.CProdN( CLocalAssum((snd (List.chop n nal)),k,t'')::nal_ta',t') in raise (Stop new_t') | _ -> anomaly (Pp.str "Not enough products.") in aux n nal_ta' in chop_n_arrow new_n t' with Stop t -> t end | _ -> anomaly (Pp.str "Not enough products.") let rec get_args b t : Constrexpr.local_binder_expr list * Constrexpr.constr_expr * Constrexpr.constr_expr = match b.CAst.v with | Constrexpr.CLambdaN (CLocalAssum(nal,k,ta) as d::rest, b') -> begin let n = List.length nal in let nal_tas,b'',t'' = get_args (CAst.make ?loc:b.CAst.loc @@ Constrexpr.CLambdaN (rest,b')) (chop_n_arrow n t) in d :: nal_tas, b'',t'' end | Constrexpr.CLambdaN ([], b) -> [],b,t | _ -> [],b,t let make_graph (f_ref : GlobRef.t) = let env = Global.env() in let sigma = Evd.from_env env in let c,c_body = match f_ref with | GlobRef.ConstRef c -> begin try c,Global.lookup_constant c with Not_found -> raise (UserError (None,str "Cannot find " ++ Printer.pr_leconstr_env env sigma (mkConst c)) ) end | _ -> raise (UserError (None, str "Not a function reference") ) in (match Global.body_of_constant_body Library.indirect_accessor c_body with | None -> error "Cannot build a graph over an axiom!" | Some (body, _, _) -> let env = Global.env () in let extern_body,extern_type = with_full_print (fun () -> (Constrextern.extern_constr false env sigma (EConstr.of_constr body), Constrextern.extern_type false env sigma (EConstr.of_constr (*FIXME*) c_body.const_type) ) ) () in let (nal_tas,b,t) = get_args extern_body extern_type in let expr_list = match b.CAst.v with | Constrexpr.CFix(l_id,fixexprl) -> let l = List.map (fun (id,recexp,bl,t,b) -> let { CAst.loc; v=rec_id } = match Option.get recexp with | { CAst.v = CStructRec id } -> id | { CAst.v = CWfRec (id,_) } -> id | { CAst.v = CMeasureRec (oid,_,_) } -> Option.get oid in let new_args = List.flatten (List.map (function | Constrexpr.CLocalDef (na,_,_)-> [] | Constrexpr.CLocalAssum (nal,_,_) -> List.map (fun {CAst.loc;v=n} -> CAst.make ?loc @@ CRef(Libnames.qualid_of_ident ?loc @@ Nameops.Name.get_id n,None)) nal | Constrexpr.CLocalPattern _ -> assert false ) nal_tas ) in let b' = add_args id.CAst.v new_args b in { Vernacexpr.fname=id; univs=None ; rec_order = Some (CAst.make (CStructRec (CAst.make rec_id))) ; binders = nal_tas@bl; rtype=t; body_def=Some b'; notations = []} ) fixexprl in l | _ -> let fname = CAst.make (Label.to_id (Constant.label c)) in [{ Vernacexpr.fname; univs=None; rec_order = None; binders=nal_tas; rtype=t; body_def=Some b; notations=[]}] in let mp = Constant.modpath c in let pstate = do_generate_principle_aux [c,Univ.Instance.empty] error_error false false expr_list in assert (Option.is_empty pstate); (* We register the infos *) List.iter (fun { Vernacexpr.fname= {CAst.v=id} } -> add_Function false (Constant.make2 mp (Label.of_id id))) expr_list) (* *************** statically typed entrypoints ************************* *) let do_generate_principle_interactive fixl : Lemmas.t = match do_generate_principle_aux [] warning_error true true fixl with | Some lemma -> lemma | None -> CErrors.anomaly (Pp.str"indfun: leaving no open proof in interactive mode") let do_generate_principle fixl : unit = match do_generate_principle_aux [] warning_error true false fixl with | Some _lemma -> CErrors.anomaly (Pp.str"indfun: leaving a goal open in non-interactive mode") | None -> ()