(************************************************************************) (* * The Coq Proof Assistant / The Coq Development Team *) (* v * Copyright INRIA, CNRS and contributors *) (* ( try match constant_opt_value_in (Global.env ()) sp with | Some c -> c | _ -> raise Not_found with Not_found -> anomaly ( str "Cannot find definition of constant " ++ Id.print (Label.to_id (Constant.label (fst sp))) ++ str "." ) ) | _ -> assert false let type_of_const sigma t = match EConstr.kind sigma t with | Const (sp, u) -> let u = EInstance.kind sigma u in (* FIXME discarding universe constraints *) Typeops.type_of_constant_in (Global.env ()) (sp, u) | _ -> assert false let constant sl s = UnivGen.constr_of_monomorphic_global (find_reference sl s) let const_of_ref = function | GlobRef.ConstRef kn -> kn | _ -> anomaly (Pp.str "ConstRef expected.") (* Generic values *) let pf_get_new_ids idl g = let ids = Tacmach.New.pf_ids_of_hyps g in let ids = Id.Set.of_list ids in List.fold_right (fun id acc -> next_global_ident_away id (Id.Set.union (Id.Set.of_list acc) ids) :: acc) idl [] let next_ident_away_in_goal ids avoid = next_ident_away_in_goal ids (Id.Set.of_list avoid) let compute_renamed_type gls id = rename_bound_vars_as_displayed (Proofview.Goal.sigma gls) (*no avoid*) Id.Set.empty (*no rels*) [] (Tacmach.New.pf_get_hyp_typ id gls) let h'_id = Id.of_string "h'" let teq_id = Id.of_string "teq" let ano_id = Id.of_string "anonymous" let x_id = Id.of_string "x" let k_id = Id.of_string "k" let v_id = Id.of_string "v" let def_id = Id.of_string "def" let p_id = Id.of_string "p" let rec_res_id = Id.of_string "rec_res" let lt = function () -> coq_init_constant "num.nat.lt" let le = function () -> Coqlib.lib_ref "num.nat.le" let ex = function () -> coq_init_constant "core.ex.type" let nat = function () -> coq_init_constant "num.nat.type" let iter_ref () = try find_reference ["Recdef"] "iter" with Not_found -> user_err Pp.(str "module Recdef not loaded") let iter_rd = function | () -> constr_of_monomorphic_global (delayed_force iter_ref) let eq = function () -> coq_init_constant "core.eq.type" let le_lt_SS = function () -> constant ["Recdef"] "le_lt_SS" let le_lt_n_Sm = function () -> coq_constant "num.nat.le_lt_n_Sm" let le_trans = function () -> coq_constant "num.nat.le_trans" let le_lt_trans = function () -> coq_constant "num.nat.le_lt_trans" let lt_S_n = function () -> coq_constant "num.nat.lt_S_n" let le_n = function () -> coq_init_constant "num.nat.le_n" let coq_sig_ref = function | () -> find_reference ["Coq"; "Init"; "Specif"] "sig" let coq_O = function () -> coq_init_constant "num.nat.O" let coq_S = function () -> coq_init_constant "num.nat.S" let lt_n_O = function () -> coq_constant "num.nat.nlt_0_r" let max_ref = function () -> find_reference ["Recdef"] "max" let max_constr = function | () -> EConstr.of_constr (constr_of_monomorphic_global (delayed_force max_ref)) let f_S t = mkApp (delayed_force coq_S, [|t|]) let rec n_x_id ids n = if Int.equal n 0 then [] else let x = next_ident_away_in_goal x_id ids in x :: n_x_id (x :: ids) (n - 1) let simpl_iter clause = reduce (Lazy { rBeta = true ; rMatch = true ; rFix = true ; rCofix = true ; rZeta = true ; rDelta = false ; rConst = [EvalConstRef (const_of_ref (delayed_force iter_ref))] }) clause (* Others ugly things ... *) let (value_f : Constr.t list -> GlobRef.t -> Constr.t) = let open Term in let open Constr in fun al fterm -> let rev_x_id_l = List.fold_left (fun x_id_l _ -> let x_id = next_ident_away_in_goal x_id x_id_l in x_id :: x_id_l) [] al in let context = List.map (fun (x, c) -> LocalAssum (make_annot (Name x) Sorts.Relevant, c)) (List.combine rev_x_id_l (List.rev al)) in let env = Environ.push_rel_context context (Global.env ()) in let glob_body = DAst.make @@ GCases ( RegularStyle , None , [ ( DAst.make @@ GApp ( DAst.make @@ GRef (fterm, None) , List.rev_map (fun x_id -> DAst.make @@ GVar x_id) rev_x_id_l ) , (Anonymous, None) ) ] , [ CAst.make ( [v_id] , [ DAst.make @@ PatCstr ( (destIndRef (delayed_force coq_sig_ref), 1) , [ DAst.make @@ PatVar (Name v_id) ; DAst.make @@ PatVar Anonymous ] , Anonymous ) ] , DAst.make @@ GVar v_id ) ] ) in let body = fst (understand env (Evd.from_env env) glob_body) (*FIXME*) in let body = EConstr.Unsafe.to_constr body in it_mkLambda_or_LetIn body context let (declare_f : Id.t -> Decls.logical_kind -> Constr.t list -> GlobRef.t -> GlobRef.t) = fun f_id kind input_type fterm_ref -> declare_fun f_id kind (value_f input_type fterm_ref) module New = struct open Tacticals.New let observe_tac = New.observe_tac ~header:(Pp.mt ()) let observe_tclTHENLIST s tacl = if do_observe () then let rec aux n = function | [] -> tclIDTAC | [tac] -> observe_tac (fun env sigma -> s env sigma ++ spc () ++ int n) tac | tac :: tacl -> observe_tac (fun env sigma -> s env sigma ++ spc () ++ int n) (tclTHEN tac (aux (succ n) tacl)) in aux 0 tacl else tclTHENLIST tacl end (* Conclusion tactics *) (* The boolean value is_mes expresses that the termination is expressed using a measure function instead of a well-founded relation. *) let tclUSER tac is_mes l = let open Tacticals.New in let clear_tac = match l with | None -> tclIDTAC | Some l -> tclMAP (fun id -> tclTRY (clear [id])) (List.rev l) in New.observe_tclTHENLIST (fun _ _ -> str "tclUSER1") [ clear_tac ; ( if is_mes then New.observe_tclTHENLIST (fun _ _ -> str "tclUSER2") [ unfold_in_concl [ ( Locus.AllOccurrences , evaluable_of_global_reference (delayed_force Indfun_common.ltof_ref) ) ] ; tac ] else tac ) ] let tclUSER_if_not_mes concl_tac is_mes names_to_suppress = if is_mes then Tacticals.New.tclCOMPLETE (Simple.apply (delayed_force well_founded_ltof)) else (* tclTHEN (Simple.apply (delayed_force acc_intro_generator_function) ) *) tclUSER concl_tac is_mes names_to_suppress (* Traveling term. Both definitions of [f_terminate] and [f_equation] use the same generic traveling mechanism. *) (* [check_not_nested forbidden e] checks that [e] does not contains any variable of [forbidden] *) let check_not_nested env sigma forbidden e = let rec check_not_nested e = match EConstr.kind sigma e with | Rel _ -> () | Int _ | Float _ -> () | Var x -> if Id.List.mem x forbidden then user_err ~hdr:"Recdef.check_not_nested" (str "check_not_nested: failure " ++ Id.print x) | Meta _ | Evar _ | Sort _ -> () | Cast (e, _, t) -> check_not_nested e; check_not_nested t | Prod (_, t, b) -> check_not_nested t; check_not_nested b | Lambda (_, t, b) -> check_not_nested t; check_not_nested b | LetIn (_, v, t, b) -> check_not_nested t; check_not_nested b; check_not_nested v | App (f, l) -> check_not_nested f | Array (_u, t, def, ty) -> Array.iter check_not_nested t; check_not_nested def; check_not_nested ty | Proj (p, c) -> check_not_nested c | Const _ -> () | Ind _ -> () | Construct _ -> () | Case (_, _, pms, (_, t), _, e, a) -> Array.iter check_not_nested pms; check_not_nested t; check_not_nested e; Array.iter (fun (_, c) -> check_not_nested c) a | Fix _ -> user_err Pp.(str "check_not_nested : Fix") | CoFix _ -> user_err Pp.(str "check_not_nested : Fix") in try check_not_nested e with UserError (_, p) -> user_err ~hdr:"_" (str "on expr : " ++ Printer.pr_leconstr_env env sigma e ++ str " " ++ p) (* ['a info] contains the local information for traveling *) type 'a infos = { nb_arg : int ; (* function number of arguments *) concl_tac : unit Proofview.tactic ; (* final tactic to finish proofs *) rec_arg_id : Id.t ; (*name of the declared recursive argument *) is_mes : bool ; (* type of recursion *) ih : Id.t ; (* induction hypothesis name *) f_id : Id.t ; (* function name *) f_constr : constr ; (* function term *) f_terminate : constr ; (* termination proof term *) func : GlobRef.t ; (* functional reference *) info : 'a ; is_main_branch : bool ; (* on the main branch or on a matched expression *) is_final : bool ; (* final first order term or not *) values_and_bounds : (Id.t * Id.t) list ; eqs : Id.t list ; forbidden_ids : Id.t list ; acc_inv : constr lazy_t ; acc_id : Id.t ; args_assoc : (constr list * constr) list } type ('a, 'b) journey_info_tac = 'a -> (* the arguments of the constructor *) 'b infos -> (* infos of the caller *) ('b infos -> unit Proofview.tactic) -> (* the continuation tactic of the caller *) 'b infos -> (* argument of the tactic *) unit Proofview.tactic (* journey_info : specifies the actions to do on the different term constructors during the traveling of the term *) type journey_info = { letiN : (Name.t * constr * types * constr, constr) journey_info_tac ; lambdA : (Name.t * types * constr, constr) journey_info_tac ; casE : ( (constr infos -> unit Proofview.tactic) -> constr infos -> unit Proofview.tactic) -> ( case_info * constr * case_invert * constr * constr array , constr ) journey_info_tac ; otherS : (unit, constr) journey_info_tac ; apP : (constr * constr list, constr) journey_info_tac ; app_reC : (constr * constr list, constr) journey_info_tac ; message : string } let add_vars sigma forbidden e = let rec aux forbidden e = match EConstr.kind sigma e with | Var x -> x :: forbidden | _ -> EConstr.fold sigma aux forbidden e in aux forbidden e let treat_case forbid_new_ids to_intros finalize_tac nb_lam e infos : unit Proofview.tactic = Proofview.Goal.enter (fun g -> let rev_context, b = decompose_lam_n (Proofview.Goal.sigma g) nb_lam e in let ids = List.fold_left (fun acc (na, _) -> let pre_id = match na.binder_name with Name x -> x | Anonymous -> ano_id in pre_id :: acc) [] rev_context in let rev_ids = pf_get_new_ids (List.rev ids) g in let new_b = substl (List.map mkVar rev_ids) b in New.observe_tclTHENLIST (fun _ _ -> str "treat_case1") [ h_intros (List.rev rev_ids) ; intro_using_then teq_id (fun _ -> Proofview.tclUNIT ()) ; Tacticals.New.onLastHypId (fun heq -> New.observe_tclTHENLIST (fun _ _ -> str "treat_case2") [ clear to_intros ; h_intros to_intros ; Proofview.Goal.enter (fun g' -> let sigma = Proofview.Goal.sigma g' in let ty_teq = Tacmach.New.pf_get_hyp_typ heq g' in let teq_lhs, teq_rhs = let _, args = try destApp sigma ty_teq with DestKO -> assert false in (args.(1), args.(2)) in let new_b' = Termops.replace_term sigma teq_lhs teq_rhs new_b in let new_infos = { infos with info = new_b' ; eqs = heq :: infos.eqs ; forbidden_ids = ( if forbid_new_ids then add_vars sigma infos.forbidden_ids new_b' else infos.forbidden_ids ) } in finalize_tac new_infos) ]) ]) let rec travel_aux jinfo continuation_tac (expr_info : constr infos) = Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in let env = Proofview.Goal.env g in match EConstr.kind sigma expr_info.info with | CoFix _ | Fix _ -> user_err Pp.(str "Function cannot treat local fixpoint or cofixpoint") | Array _ -> user_err Pp.(str "Function cannot treat arrays") | Proj _ -> user_err Pp.(str "Function cannot treat projections") | LetIn (na, b, t, e) -> let new_continuation_tac = jinfo.letiN (na.binder_name, b, t, e) expr_info continuation_tac in travel jinfo new_continuation_tac {expr_info with info = b; is_final = false} | Rel _ -> anomaly (Pp.str "Free var in goal conclusion!") | Prod _ -> ( try check_not_nested env sigma (expr_info.f_id :: expr_info.forbidden_ids) expr_info.info; jinfo.otherS () expr_info continuation_tac expr_info with e when CErrors.noncritical e -> user_err ~hdr:"Recdef.travel" ( str "the term " ++ Printer.pr_leconstr_env env sigma expr_info.info ++ str " can not contain a recursive call to " ++ Id.print expr_info.f_id ) ) | Lambda (n, t, b) -> ( try check_not_nested env sigma (expr_info.f_id :: expr_info.forbidden_ids) expr_info.info; jinfo.otherS () expr_info continuation_tac expr_info with e when CErrors.noncritical e -> user_err ~hdr:"Recdef.travel" ( str "the term " ++ Printer.pr_leconstr_env env sigma expr_info.info ++ str " can not contain a recursive call to " ++ Id.print expr_info.f_id ) ) | Case (ci, u, pms, t, iv, a, l) -> let (ci, t, iv, a, l) = EConstr.expand_case env sigma (ci, u, pms, t, iv, a, l) in let continuation_tac_a = jinfo.casE (travel jinfo) (ci, t, iv, a, l) expr_info continuation_tac in travel jinfo continuation_tac_a {expr_info with info = a; is_main_branch = false; is_final = false} | App _ -> ( let f, args = decompose_app sigma expr_info.info in if EConstr.eq_constr sigma f expr_info.f_constr then jinfo.app_reC (f, args) expr_info continuation_tac expr_info else match EConstr.kind sigma f with | App _ -> assert false (* f is coming from a decompose_app *) | Const _ | Construct _ | Rel _ | Evar _ | Meta _ | Ind _ | Sort _ |Prod _ | Var _ -> let new_infos = {expr_info with info = (f, args)} in let new_continuation_tac = jinfo.apP (f, args) expr_info continuation_tac in travel_args jinfo expr_info.is_main_branch new_continuation_tac new_infos | Case _ -> user_err ~hdr:"Recdef.travel" ( str "the term " ++ Printer.pr_leconstr_env env sigma expr_info.info ++ str " can not contain an applied match (See Limitation in \ Section 2.3 of refman)" ) | _ -> anomaly ( Pp.str "travel_aux : unexpected " ++ Printer.pr_leconstr_env env sigma expr_info.info ++ Pp.str "." ) ) | Cast (t, _, _) -> travel jinfo continuation_tac {expr_info with info = t} | Const _ | Var _ | Meta _ | Evar _ | Sort _ | Construct _ | Ind _ |Int _ | Float _ -> let new_continuation_tac = jinfo.otherS () expr_info continuation_tac in new_continuation_tac expr_info) and travel_args jinfo is_final continuation_tac infos = let f_args', args = infos.info in match args with | [] -> continuation_tac {infos with info = f_args'; is_final} | arg :: args' -> let new_continuation_tac new_infos = let new_arg = new_infos.info in travel_args jinfo is_final continuation_tac {new_infos with info = (mkApp (f_args', [|new_arg|]), args')} in travel jinfo new_continuation_tac {infos with info = arg; is_final = false} and travel jinfo continuation_tac expr_info = New.observe_tac (fun env sigma -> str jinfo.message ++ Printer.pr_leconstr_env env sigma expr_info.info) (travel_aux jinfo continuation_tac expr_info) (* Termination proof *) let rec prove_lt hyple = Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in try let varx, varz = match decompose_app sigma (Proofview.Goal.concl g) with | _, x :: z :: _ when isVar sigma x && isVar sigma z -> (x, z) | _ -> assert false in let h = List.find (fun id -> match decompose_app sigma (Tacmach.New.pf_get_hyp_typ id g) with | _, t :: _ -> EConstr.eq_constr sigma t varx | _ -> false) hyple in let y = List.hd (List.tl (snd (decompose_app sigma (Tacmach.New.pf_get_hyp_typ h g)))) in New.observe_tclTHENLIST (fun _ _ -> str "prove_lt1") [ apply (mkApp (le_lt_trans (), [|varx; y; varz; mkVar h|])) ; New.observe_tac (fun _ _ -> str "prove_lt") (prove_lt hyple) ] with Not_found -> New.observe_tclTHENLIST (fun _ _ -> str "prove_lt2") [ apply (delayed_force lt_S_n) ; New.observe_tac (fun _ _ -> str "assumption: " ++ Printer.pr_goal Evd.{it = Proofview.Goal.goal g; sigma}) assumption ]) let rec destruct_bounds_aux infos (bound, hyple, rechyps) lbounds = let open Tacticals.New in Proofview.Goal.enter (fun g -> match lbounds with | [] -> let ids = Tacmach.New.pf_ids_of_hyps g in let s_max = mkApp (delayed_force coq_S, [|bound|]) in let k = next_ident_away_in_goal k_id ids in let ids = k :: ids in let h' = next_ident_away_in_goal h'_id ids in let ids = h' :: ids in let def = next_ident_away_in_goal def_id ids in New.observe_tclTHENLIST (fun _ _ -> str "destruct_bounds_aux1") [ split (ImplicitBindings [s_max]) ; intro_then (fun id -> New.observe_tac (fun _ _ -> str "destruct_bounds_aux") (tclTHENS (simplest_case (mkVar id)) [ New.observe_tclTHENLIST (fun _ _ -> str "") [ intro_using_then h_id (* We don't care about the refreshed name, accessed only through auto? *) (fun _ -> Proofview.tclUNIT ()) ; simplest_elim (mkApp (delayed_force lt_n_O, [|s_max|])) ; default_full_auto ] ; New.observe_tclTHENLIST (fun _ _ -> str "destruct_bounds_aux2") [ New.observe_tac (fun _ _ -> str "clearing k ") (clear [id]) ; h_intros [k; h'; def] ; New.observe_tac (fun _ _ -> str "simple_iter") (simpl_iter Locusops.onConcl) ; New.observe_tac (fun _ _ -> str "unfold functional") (unfold_in_concl [ ( Locus.OnlyOccurrences [1] , evaluable_of_global_reference infos.func ) ]) ; New.observe_tclTHENLIST (fun _ _ -> str "test") [ list_rewrite true (List.fold_right (fun e acc -> (mkVar e, true) :: acc) infos.eqs (List.map (fun e -> (e, true)) rechyps)) ; (* list_rewrite true *) (* (List.map (fun e -> (mkVar e,true)) infos.eqs) *) (* ; *) New.observe_tac (fun _ _ -> str "finishing") (tclORELSE intros_reflexivity (New.observe_tac (fun _ _ -> str "calling prove_lt") (prove_lt hyple))) ] ] ])) ] | (_, v_bound) :: l -> New.observe_tclTHENLIST (fun _ _ -> str "destruct_bounds_aux3") [ simplest_elim (mkVar v_bound) ; clear [v_bound] ; tclDO 2 intro ; onNthHypId 1 (fun p_hyp -> onNthHypId 2 (fun p -> New.observe_tclTHENLIST (fun _ _ -> str "destruct_bounds_aux4") [ simplest_elim (mkApp (delayed_force max_constr, [|bound; mkVar p|])) ; tclDO 3 intro ; onNLastHypsId 3 (fun lids -> match lids with | [hle2; hle1; pmax] -> destruct_bounds_aux infos ( mkVar pmax , hle1 :: hle2 :: hyple , mkVar p_hyp :: rechyps ) l | _ -> assert false) ])) ]) let destruct_bounds infos = destruct_bounds_aux infos (delayed_force coq_O, [], []) infos.values_and_bounds let terminate_app f_and_args expr_info continuation_tac infos = if expr_info.is_final && expr_info.is_main_branch then New.observe_tclTHENLIST (fun _ _ -> str "terminate_app1") [ continuation_tac infos ; New.observe_tac (fun _ _ -> str "first split") (split (ImplicitBindings [infos.info])) ; New.observe_tac (fun _ _ -> str "destruct_bounds (1)") (destruct_bounds infos) ] else continuation_tac infos let terminate_others _ expr_info continuation_tac infos = if expr_info.is_final && expr_info.is_main_branch then New.observe_tclTHENLIST (fun _ _ -> str "terminate_others") [ continuation_tac infos ; New.observe_tac (fun _ _ -> str "first split") (split (ImplicitBindings [infos.info])) ; New.observe_tac (fun _ _ -> str "destruct_bounds") (destruct_bounds infos) ] else continuation_tac infos let terminate_letin (na, b, t, e) expr_info continuation_tac info = Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in let env = Proofview.Goal.env g in let new_e = subst1 info.info e in let new_forbidden = let forbid = try check_not_nested env sigma (expr_info.f_id :: expr_info.forbidden_ids) b; true with e when CErrors.noncritical e -> false in if forbid then match na with | Anonymous -> info.forbidden_ids | Name id -> id :: info.forbidden_ids else info.forbidden_ids in continuation_tac {info with info = new_e; forbidden_ids = new_forbidden}) let pf_type c tac = let open Tacticals.New in Proofview.Goal.enter (fun gl -> let env = Proofview.Goal.env gl in let sigma = Proofview.Goal.sigma gl in let evars, ty = Typing.type_of env sigma c in tclTHEN (Proofview.Unsafe.tclEVARS evars) (tac ty)) let pf_typel l tac = let rec aux tys l = match l with | [] -> tac (List.rev tys) | hd :: tl -> pf_type hd (fun ty -> aux (ty :: tys) tl) in aux [] l (* This is like the previous one except that it also rewrite on all hypotheses except the ones given in the first argument. All the modified hypotheses are generalized in the process and should be introduced back later; the result is the pair of the tactic and the list of hypotheses that have been generalized and cleared. *) let mkDestructEq not_on_hyp env sigma expr = let hyps = EConstr.named_context env in let to_revert = Util.List.map_filter (fun decl -> let open Context.Named.Declaration in let id = get_id decl in if Id.List.mem id not_on_hyp || not (Termops.dependent sigma expr (get_type decl)) then None else Some id) hyps in let to_revert_constr = List.rev_map mkVar to_revert in let sigma, type_of_expr = Typing.type_of env sigma expr in let new_hyps = mkApp (Lazy.force refl_equal, [|type_of_expr; expr|]) :: to_revert_constr in let tac = pf_typel new_hyps (fun _ -> New.observe_tclTHENLIST (fun _ _ -> str "mkDestructEq") [ generalize new_hyps ; Proofview.Goal.enter (fun g2 -> let changefun patvars env sigma = pattern_occs [(Locus.AllOccurrencesBut [1], expr)] (Proofview.Goal.env g2) sigma (Proofview.Goal.concl g2) in change_in_concl ~check:true None changefun) ; simplest_case expr ]) in (sigma, tac, to_revert) let terminate_case next_step (ci, a, iv, t, l) expr_info continuation_tac infos = let open Tacticals.New in Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in let env = Proofview.Goal.env g in let f_is_present = try check_not_nested env sigma (expr_info.f_id :: expr_info.forbidden_ids) a; false with e when CErrors.noncritical e -> true in let a' = infos.info in let new_info = { infos with info = mkCase (EConstr.contract_case env sigma (ci, a, iv, a', l)) ; is_main_branch = expr_info.is_main_branch ; is_final = expr_info.is_final } in let sigma, destruct_tac, rev_to_thin_intro = mkDestructEq [expr_info.rec_arg_id] env sigma a' in let to_thin_intro = List.rev rev_to_thin_intro in New.observe_tac (fun _ _ -> str "treating cases (" ++ int (Array.length l) ++ str ")" ++ spc () ++ Printer.pr_leconstr_env env sigma a') ( try tclTHENS destruct_tac (List.map_i (fun i e -> New.observe_tac (fun _ _ -> str "do treat case") (treat_case f_is_present to_thin_intro (next_step continuation_tac) ci.ci_cstr_ndecls.(i) e new_info)) 0 (Array.to_list l)) with | UserError (Some "Refiner.thensn_tac3", _) |UserError (Some "Refiner.tclFAIL_s", _) -> New.observe_tac (fun _ _ -> str "is computable " ++ Printer.pr_leconstr_env env sigma new_info.info) (next_step continuation_tac { new_info with info = Reductionops.nf_betaiotazeta env sigma new_info.info }) )) let terminate_app_rec (f, args) expr_info continuation_tac _ = let open Tacticals.New in Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in let env = Proofview.Goal.env g in List.iter (check_not_nested env sigma (expr_info.f_id :: expr_info.forbidden_ids)) args; try let v = List.assoc_f (List.equal (EConstr.eq_constr sigma)) args expr_info.args_assoc in let new_infos = {expr_info with info = v} in New.observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec") [ continuation_tac new_infos ; ( if expr_info.is_final && expr_info.is_main_branch then New.observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec1") [ New.observe_tac (fun _ _ -> str "first split") (split (ImplicitBindings [new_infos.info])) ; New.observe_tac (fun _ _ -> str "destruct_bounds (3)") (destruct_bounds new_infos) ] else Proofview.tclUNIT () ) ] with Not_found -> New.observe_tac (fun _ _ -> str "terminate_app_rec not found") (tclTHENS (simplest_elim (mkApp (mkVar expr_info.ih, Array.of_list args))) [ New.observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec2") [ intro_using_then rec_res_id (* refreshed name gotten from onNthHypId *) (fun _ -> Proofview.tclUNIT ()) ; intro ; onNthHypId 1 (fun v_bound -> onNthHypId 2 (fun v -> let new_infos = { expr_info with info = mkVar v ; values_and_bounds = (v, v_bound) :: expr_info.values_and_bounds ; args_assoc = (args, mkVar v) :: expr_info.args_assoc } in New.observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec3") [ continuation_tac new_infos ; ( if expr_info.is_final && expr_info.is_main_branch then New.observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec4") [ New.observe_tac (fun _ _ -> str "first split") (split (ImplicitBindings [new_infos.info])) ; New.observe_tac (fun _ _ -> str "destruct_bounds (2)") (destruct_bounds new_infos) ] else Proofview.tclUNIT () ) ])) ] ; New.observe_tac (fun _ _ -> str "proving decreasing") (tclTHENS (* proof of args < formal args *) (apply (Lazy.force expr_info.acc_inv)) [ New.observe_tac (fun _ _ -> str "assumption") assumption ; New.observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec5") [ tclTRY (list_rewrite true (List.map (fun e -> (mkVar e, true)) expr_info.eqs)) ; tclUSER expr_info.concl_tac true (Some ( expr_info.ih :: expr_info.acc_id :: (fun (x, y) -> y) (List.split expr_info.values_and_bounds) )) ] ]) ])) let terminate_info = { message = "prove_terminate with term " ; letiN = terminate_letin ; lambdA = (fun _ _ _ _ -> assert false) ; casE = terminate_case ; otherS = terminate_others ; apP = terminate_app ; app_reC = terminate_app_rec } let prove_terminate = travel terminate_info (* Equation proof *) let equation_case next_step case expr_info continuation_tac infos = New.observe_tac (fun _ _ -> str "equation case") (terminate_case next_step case expr_info continuation_tac infos) let rec prove_le () = let open Tacticals.New in Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in let x, z = let _, args = decompose_app sigma (Proofview.Goal.concl g) in (List.hd args, List.hd (List.tl args)) in tclFIRST [ assumption ; apply (delayed_force le_n) ; begin try let matching_fun c = match EConstr.kind sigma c with | App (c, [|x0; _|]) -> EConstr.isVar sigma x0 && Id.equal (destVar sigma x0) (destVar sigma x) && EConstr.isRefX sigma (le ()) c | _ -> false in let h, t = List.find (fun (_, t) -> matching_fun t) (Tacmach.New.pf_hyps_types g) in let y = let _, args = decompose_app sigma t in List.hd (List.tl args) in New.observe_tclTHENLIST (fun _ _ -> str "prove_le") [ apply (mkApp (le_trans (), [|x; y; z; mkVar h|])) ; New.observe_tac (fun _ _ -> str "prove_le (rec)") (prove_le ()) ] with Not_found -> Tacticals.New.tclFAIL 0 (mt ()) end ]) let rec make_rewrite_list expr_info max = function | [] -> Proofview.tclUNIT () | (_, p, hp) :: l -> let open Tacticals.New in New.observe_tac (fun _ _ -> str "make_rewrite_list") (tclTHENS (New.observe_tac (fun _ _ -> str "rewrite heq on " ++ Id.print p) (Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in let t_eq = compute_renamed_type g hp in let k, def = let k_na, _, t = destProd sigma t_eq in let _, _, t = destProd sigma t in let def_na, _, _ = destProd sigma t in ( Nameops.Name.get_id k_na.binder_name , Nameops.Name.get_id def_na.binder_name ) in general_rewrite_bindings false Locus.AllOccurrences true (* dep proofs also: *) true ( mkVar hp , ExplicitBindings [ CAst.make @@ (NamedHyp def, expr_info.f_constr) ; CAst.make @@ (NamedHyp k, f_S max) ] ) false))) [ make_rewrite_list expr_info max l ; New.observe_tclTHENLIST (fun _ _ -> str "make_rewrite_list") [ (* x < S max proof *) apply (delayed_force le_lt_n_Sm) ; New.observe_tac (fun _ _ -> str "prove_le(2)") (prove_le ()) ] ]) let make_rewrite expr_info l hp max = let open Tacticals.New in tclTHENFIRST (New.observe_tac (fun _ _ -> str "make_rewrite") (make_rewrite_list expr_info max l)) (New.observe_tac (fun _ _ -> str "make_rewrite") (tclTHENS (Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in let t_eq = compute_renamed_type g hp in let k, def = let k_na, _, t = destProd sigma t_eq in let _, _, t = destProd sigma t in let def_na, _, _ = destProd sigma t in ( Nameops.Name.get_id k_na.binder_name , Nameops.Name.get_id def_na.binder_name ) in New.observe_tac (fun _ _ -> str "general_rewrite_bindings") (general_rewrite_bindings false Locus.AllOccurrences true (* dep proofs also: *) true ( mkVar hp , ExplicitBindings [ CAst.make @@ (NamedHyp def, expr_info.f_constr) ; CAst.make @@ (NamedHyp k, f_S (f_S max)) ] ) false))) [ New.observe_tac (fun _ _ -> str "make_rewrite finalize") ((* tclORELSE( h_reflexivity) *) New.observe_tclTHENLIST (fun _ _ -> str "make_rewrite") [ simpl_iter Locusops.onConcl ; New.observe_tac (fun _ _ -> str "unfold functional") (unfold_in_concl [ ( Locus.OnlyOccurrences [1] , evaluable_of_global_reference expr_info.func ) ]) ; list_rewrite true (List.map (fun e -> (mkVar e, true)) expr_info.eqs) ; New.observe_tac (fun _ _ -> str "h_reflexivity") intros_reflexivity ]) ; New.observe_tclTHENLIST (fun _ _ -> str "make_rewrite1") [ (* x < S (S max) proof *) apply (EConstr.of_constr (delayed_force le_lt_SS)) ; New.observe_tac (fun _ _ -> str "prove_le (3)") (prove_le ()) ] ])) let rec compute_max rew_tac max l = match l with | [] -> rew_tac max | (_, p, _) :: l -> let open Tacticals.New in New.observe_tclTHENLIST (fun _ _ -> str "compute_max") [ simplest_elim (mkApp (delayed_force max_constr, [|max; mkVar p|])) ; tclDO 3 intro ; onNLastHypsId 3 (fun lids -> match lids with | [hle2; hle1; pmax] -> compute_max rew_tac (mkVar pmax) l | _ -> assert false) ] let rec destruct_hex expr_info acc l = let open Tacticals.New in match l with | [] -> ( match List.rev acc with | [] -> Proofview.tclUNIT () | (_, p, hp) :: tl -> New.observe_tac (fun _ _ -> str "compute max ") (compute_max (make_rewrite expr_info tl hp) (mkVar p) tl) ) | (v, hex) :: l -> New.observe_tclTHENLIST (fun _ _ -> str "destruct_hex") [ simplest_case (mkVar hex) ; clear [hex] ; tclDO 2 intro ; onNthHypId 1 (fun hp -> onNthHypId 2 (fun p -> New.observe_tac (fun _ _ -> str "destruct_hex after " ++ Id.print hp ++ spc () ++ Id.print p) (destruct_hex expr_info ((v, p, hp) :: acc) l))) ] let rec intros_values_eq expr_info acc = let open Tacticals.New in tclORELSE (New.observe_tclTHENLIST (fun _ _ -> str "intros_values_eq") [ tclDO 2 intro ; onNthHypId 1 (fun hex -> onNthHypId 2 (fun v -> intros_values_eq expr_info ((v, hex) :: acc))) ]) (tclCOMPLETE (destruct_hex expr_info [] acc)) let equation_others _ expr_info continuation_tac infos = let open Tacticals.New in if expr_info.is_final && expr_info.is_main_branch then New.observe_tac (fun env sigma -> str "equation_others (cont_tac +intros) " ++ Printer.pr_leconstr_env env sigma expr_info.info) (tclTHEN (continuation_tac infos) (New.observe_tac (fun env sigma -> str "intros_values_eq equation_others " ++ Printer.pr_leconstr_env env sigma expr_info.info) (intros_values_eq expr_info []))) else New.observe_tac (fun env sigma -> str "equation_others (cont_tac) " ++ Printer.pr_leconstr_env env sigma expr_info.info) (continuation_tac infos) let equation_app f_and_args expr_info continuation_tac infos = if expr_info.is_final && expr_info.is_main_branch then New.observe_tac (fun _ _ -> str "intros_values_eq equation_app") (intros_values_eq expr_info []) else continuation_tac infos let equation_app_rec (f, args) expr_info continuation_tac info = Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in try let v = List.assoc_f (List.equal (EConstr.eq_constr sigma)) args expr_info.args_assoc in let new_infos = {expr_info with info = v} in New.observe_tac (fun _ _ -> str "app_rec found") (continuation_tac new_infos) with Not_found -> if expr_info.is_final && expr_info.is_main_branch then New.observe_tclTHENLIST (fun _ _ -> str "equation_app_rec") [ simplest_case (mkApp (expr_info.f_terminate, Array.of_list args)) ; continuation_tac { expr_info with args_assoc = (args, delayed_force coq_O) :: expr_info.args_assoc } ; New.observe_tac (fun _ _ -> str "app_rec intros_values_eq") (intros_values_eq expr_info []) ] else New.observe_tclTHENLIST (fun _ _ -> str "equation_app_rec1") [ simplest_case (mkApp (expr_info.f_terminate, Array.of_list args)) ; New.observe_tac (fun _ _ -> str "app_rec not_found") (continuation_tac { expr_info with args_assoc = (args, delayed_force coq_O) :: expr_info.args_assoc }) ]) let equation_info = { message = "prove_equation with term " ; letiN = (fun _ -> assert false) ; lambdA = (fun _ _ _ _ -> assert false) ; casE = equation_case ; otherS = equation_others ; apP = equation_app ; app_reC = equation_app_rec } let prove_eq = travel equation_info (* wrappers *) (* [compute_terminate_type] computes the type of the Definition f_terminate from the type of f_F *) let compute_terminate_type nb_args func = let open Term in let open Constr in let open CVars in let _, a_arrow_b, _ = destLambda (def_of_const (constr_of_monomorphic_global func)) in let rev_args, b = decompose_prod_n nb_args a_arrow_b in let left = mkApp ( delayed_force iter_rd , Array.of_list ( lift 5 a_arrow_b :: mkRel 3 :: constr_of_monomorphic_global func :: mkRel 1 :: List.rev (List.map_i (fun i _ -> mkRel (6 + i)) 0 rev_args) ) ) in let right = mkRel 5 in let delayed_force c = EConstr.Unsafe.to_constr (delayed_force c) in let equality = mkApp (delayed_force eq, [|lift 5 b; left; right|]) in let result = mkProd (make_annot (Name def_id) Sorts.Relevant, lift 4 a_arrow_b, equality) in let cond = mkApp (delayed_force lt, [|mkRel 2; mkRel 1|]) in let nb_iter = mkApp ( delayed_force ex , [| delayed_force nat ; mkLambda ( make_annot (Name p_id) Sorts.Relevant , delayed_force nat , mkProd ( make_annot (Name k_id) Sorts.Relevant , delayed_force nat , mkArrow cond Sorts.Relevant result ) ) |] ) in let value = mkApp ( constr_of_monomorphic_global (Util.delayed_force coq_sig_ref) , [|b; mkLambda (make_annot (Name v_id) Sorts.Relevant, b, nb_iter)|] ) in compose_prod rev_args value let termination_proof_header is_mes input_type ids args_id relation rec_arg_num rec_arg_id tac wf_tac : unit Proofview.tactic = let open Tacticals.New in Proofview.Goal.enter (fun g -> let nargs = List.length args_id in let pre_rec_args = List.rev_map mkVar (fst (List.chop (rec_arg_num - 1) args_id)) in let relation = substl pre_rec_args relation in let input_type = substl pre_rec_args input_type in let wf_thm = next_ident_away_in_goal (Id.of_string "wf_R") ids in let wf_rec_arg = next_ident_away_in_goal (Id.of_string ("Acc_" ^ Id.to_string rec_arg_id)) (wf_thm :: ids) in let hrec = next_ident_away_in_goal hrec_id (wf_rec_arg :: wf_thm :: ids) in let acc_inv = lazy (mkApp ( delayed_force acc_inv_id , [|input_type; relation; mkVar rec_arg_id|] )) in tclTHEN (h_intros args_id) (tclTHENS (New.observe_tac (fun _ _ -> str "first assert") (assert_before (Name wf_rec_arg) (mkApp ( delayed_force acc_rel , [|input_type; relation; mkVar rec_arg_id|] )))) [ (* accesibility proof *) tclTHENS (New.observe_tac (fun _ _ -> str "second assert") (assert_before (Name wf_thm) (mkApp (delayed_force well_founded, [|input_type; relation|])))) [ (* interactive proof that the relation is well_founded *) New.observe_tac (fun _ _ -> str "wf_tac") (wf_tac is_mes (Some args_id)) ; (* this gives the accessibility argument *) New.observe_tac (fun _ _ -> str "apply wf_thm") (Simple.apply (mkApp (mkVar wf_thm, [|mkVar rec_arg_id|]))) ] ; (* rest of the proof *) New.observe_tclTHENLIST (fun _ _ -> str "rest of proof") [ New.observe_tac (fun _ _ -> str "generalize") (onNLastHypsId (nargs + 1) (tclMAP (fun id -> tclTHEN (Tactics.generalize [mkVar id]) (clear [id])))) ; New.observe_tac (fun _ _ -> str "fix") (fix hrec (nargs + 1)) ; h_intros args_id ; Simple.intro wf_rec_arg ; New.observe_tac (fun _ _ -> str "tac") (tac wf_rec_arg hrec wf_rec_arg acc_inv) ] ])) let rec instantiate_lambda sigma t l = match l with | [] -> t | a :: l -> let _, _, body = destLambda sigma t in instantiate_lambda sigma (subst1 a body) l let whole_start concl_tac nb_args is_mes func input_type relation rec_arg_num : unit Proofview.tactic = Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in let hyps = Proofview.Goal.hyps g in let ids = Termops.ids_of_named_context hyps in let func_body = def_of_const (constr_of_monomorphic_global func) in let func_body = EConstr.of_constr func_body in let f_name, _, body1 = destLambda sigma func_body in let f_id = match f_name.binder_name with | Name f_id -> next_ident_away_in_goal f_id ids | Anonymous -> anomaly (Pp.str "Anonymous function.") in let n_names_types, _ = decompose_lam_n sigma nb_args body1 in let n_ids, ids = List.fold_left (fun (n_ids, ids) (n_name, _) -> match n_name.binder_name with | Name id -> let n_id = next_ident_away_in_goal id ids in (n_id :: n_ids, n_id :: ids) | _ -> anomaly (Pp.str "anonymous argument.")) ([], f_id :: ids) n_names_types in let rec_arg_id = List.nth n_ids (rec_arg_num - 1) in let expr = instantiate_lambda sigma func_body (mkVar f_id :: List.map mkVar n_ids) in termination_proof_header is_mes input_type ids n_ids relation rec_arg_num rec_arg_id (fun rec_arg_id hrec acc_id acc_inv -> prove_terminate (fun infos -> Proofview.tclUNIT ()) { is_main_branch = true ; (* we are on the main branche (i.e. still on a match ... with .... end *) is_final = true ; (* and on leaf (more or less) *) f_terminate = delayed_force coq_O ; nb_arg = nb_args ; concl_tac ; rec_arg_id ; is_mes ; ih = hrec ; f_id ; f_constr = mkVar f_id ; func ; info = expr ; acc_inv ; acc_id ; values_and_bounds = [] ; eqs = [] ; forbidden_ids = [] ; args_assoc = [] }) (fun b ids -> tclUSER_if_not_mes concl_tac b ids)) let get_current_subgoals_types pstate = let p = Declare.Proof.get pstate in let Proof.{goals = sgs; sigma; _} = Proof.data p in (sigma, List.map (Goal.V82.abstract_type sigma) sgs) exception EmptySubgoals let build_and_l sigma l = let and_constr = UnivGen.constr_of_monomorphic_global @@ Coqlib.lib_ref "core.and.type" in let conj_constr = Coqlib.lib_ref "core.and.conj" in let mk_and p1 p2 = mkApp (EConstr.of_constr and_constr, [|p1; p2|]) in let rec is_well_founded t = match EConstr.kind sigma t with | Prod (_, _, t') -> is_well_founded t' | App (_, _) -> let f, _ = decompose_app sigma t in EConstr.eq_constr sigma f (well_founded ()) | _ -> false in let compare t1 t2 = let b1, b2 = (is_well_founded t1, is_well_founded t2) in if (b1 && b2) || not (b1 || b2) then 0 else if b1 && not b2 then 1 else -1 in let l = List.sort compare l in let rec f = function | [] -> raise EmptySubgoals | [p] -> (p, tclIDTAC, 1) | p1 :: pl -> let c, tac, nb = f pl in ( mk_and p1 c , tclTHENS (apply (EConstr.of_constr (constr_of_monomorphic_global conj_constr))) [tclIDTAC; tac] , nb + 1 ) in f l let is_rec_res id = let rec_res_name = Id.to_string rec_res_id in let id_name = Id.to_string id in try String.equal (String.sub id_name 0 (String.length rec_res_name)) rec_res_name with Invalid_argument _ -> false let clear_goals sigma = let rec clear_goal t = match EConstr.kind sigma t with | Prod (({binder_name = Name id} as na), t', b) -> let b' = clear_goal b in if noccurn sigma 1 b' && is_rec_res id then Vars.lift (-1) b' else if b' == b then t else mkProd (na, t', b') | _ -> EConstr.map sigma clear_goal t in List.map clear_goal let build_new_goal_type lemma = let sigma, sub_gls_types = get_current_subgoals_types lemma in (* Pp.msgnl (str "sub_gls_types1 := " ++ Util.prlist_with_sep (fun () -> Pp.fnl () ++ Pp.fnl ()) Printer.pr_lconstr sub_gls_types); *) let sub_gls_types = clear_goals sigma sub_gls_types in (* Pp.msgnl (str "sub_gls_types2 := " ++ Pp.prlist_with_sep (fun () -> Pp.fnl () ++ Pp.fnl ()) Printer.pr_lconstr sub_gls_types); *) let res = build_and_l sigma sub_gls_types in (sigma, res) let is_opaque_constant c = let cb = Global.lookup_constant c in let open Vernacexpr in match cb.Declarations.const_body with | Declarations.OpaqueDef _ -> Opaque | Declarations.Undef _ -> Opaque | Declarations.Def _ -> Transparent | Declarations.Primitive _ -> Opaque let open_new_goal ~lemma build_proof sigma using_lemmas ref_ goal_name (gls_type, decompose_and_tac, nb_goal) = (* Pp.msgnl (str "gls_type := " ++ Printer.pr_lconstr gls_type); *) let current_proof_name = Declare.Proof.get_name lemma in let name = match goal_name with | Some s -> s | None -> ( try add_suffix current_proof_name "_subproof" with e when CErrors.noncritical e -> anomaly (Pp.str "open_new_goal with an unnamed theorem.") ) in let na = next_global_ident_away name Id.Set.empty in if Termops.occur_existential sigma gls_type then CErrors.user_err Pp.(str "\"abstract\" cannot handle existentials"); let hook _ = let opacity = let na_ref = qualid_of_ident na in let na_global = Smartlocate.global_with_alias na_ref in match na_global with | GlobRef.ConstRef c -> is_opaque_constant c | _ -> anomaly ~label:"equation_lemma" (Pp.str "not a constant.") in let lemma = mkConst (Names.Constant.make1 (Lib.make_kn na)) in ref_ := Value (EConstr.Unsafe.to_constr lemma); let lid = ref [] in let h_num = ref (-1) in let env = Global.env () in let start_tac = let open Tacmach.New in let open Tacticals.New in Proofview.Goal.enter (fun gl -> let hid = next_ident_away_in_goal h_id (pf_ids_of_hyps gl) in New.observe_tclTHENLIST (fun _ _ -> mt ()) [ generalize [lemma] ; Simple.intro hid ; Proofview.Goal.enter (fun gl -> let ids = pf_ids_of_hyps gl in tclTHEN (Elim.h_decompose_and (mkVar hid)) (Proofview.Goal.enter (fun gl -> let ids' = pf_ids_of_hyps gl in lid := List.rev (List.subtract Id.equal ids' ids); if List.is_empty !lid then lid := [hid]; tclIDTAC))) ]) in let end_tac = let open Tacmach.New in let open Tacticals.New in Proofview.Goal.enter (fun gl -> let sigma = project gl in match EConstr.kind sigma (pf_concl gl) with | App (f, _) when EConstr.eq_constr sigma f (well_founded ()) -> Auto.h_auto None [] (Some []) | _ -> incr h_num; tclCOMPLETE (tclFIRST [ tclTHEN (eapply_with_bindings (mkVar (List.nth !lid !h_num), NoBindings)) e_assumption ; Eauto.eauto_with_bases (true, 5) [(fun _ sigma -> (sigma, Lazy.force refl_equal))] [Hints.Hint_db.empty TransparentState.empty false] ])) in let lemma = build_proof env (Evd.from_env env) start_tac end_tac in let (_ : _ list) = Declare.Proof.save_regular ~proof:lemma ~opaque:opacity ~idopt:None in () in let info = Declare.Info.make ~hook:(Declare.Hook.make hook) () in let cinfo = Declare.CInfo.make ~name:na ~typ:gls_type () in let lemma = Declare.Proof.start ~cinfo ~info sigma in let lemma = if Indfun_common.is_strict_tcc () then fst @@ Declare.Proof.by tclIDTAC lemma else fst @@ Declare.Proof.by (tclTHEN decompose_and_tac (tclORELSE (tclFIRST (List.map (fun c -> Tacticals.New.tclTHENLIST [ intros ; Simple.apply (fst (interp_constr (Global.env ()) Evd.empty c)) (*FIXME*) ; Tacticals.New.tclCOMPLETE Auto.default_auto ]) using_lemmas)) tclIDTAC)) lemma in if Declare.Proof.get_open_goals lemma = 0 then (defined lemma; None) else Some lemma let com_terminate interactive_proof tcc_lemma_name tcc_lemma_ref is_mes fonctional_ref input_type relation rec_arg_num thm_name using_lemmas nb_args ctx hook = let start_proof env ctx tac_start tac_end = let cinfo = Declare.CInfo.make ~name:thm_name ~typ:(EConstr.of_constr (compute_terminate_type nb_args fonctional_ref)) () in let info = Declare.Info.make ~hook () in let lemma = Declare.Proof.start ~cinfo ~info ctx in let lemma = fst @@ Declare.Proof.by (New.observe_tac (fun _ _ -> str "starting_tac") tac_start) lemma in fst @@ Declare.Proof.by (New.observe_tac (fun _ _ -> str "whole_start") (whole_start tac_end nb_args is_mes fonctional_ref input_type relation rec_arg_num)) lemma in let lemma = start_proof Global.(env ()) ctx Tacticals.New.tclIDTAC Tacticals.New.tclIDTAC in try let sigma, new_goal_type = build_new_goal_type lemma in let sigma = Evd.from_ctx (Evd.evar_universe_context sigma) in open_new_goal ~lemma start_proof sigma using_lemmas tcc_lemma_ref (Some tcc_lemma_name) new_goal_type with EmptySubgoals -> (* a non recursive function declared with measure ! *) tcc_lemma_ref := Not_needed; if interactive_proof then Some lemma else (defined lemma; None) let start_equation (f : GlobRef.t) (term_f : GlobRef.t) (cont_tactic : Id.t list -> unit Proofview.tactic) = Proofview.Goal.enter (fun g -> let sigma = Proofview.Goal.sigma g in let ids = Tacmach.New.pf_ids_of_hyps g in let terminate_constr = constr_of_monomorphic_global term_f in let terminate_constr = EConstr.of_constr terminate_constr in let nargs = nb_prod sigma (EConstr.of_constr (type_of_const sigma terminate_constr)) in let x = n_x_id ids nargs in New.observe_tac (fun _ _ -> str "start_equation") (New.observe_tclTHENLIST (fun _ _ -> str "start_equation") [ h_intros x ; unfold_in_concl [(Locus.AllOccurrences, evaluable_of_global_reference f)] ; New.observe_tac (fun _ _ -> str "simplest_case") (simplest_case (mkApp (terminate_constr, Array.of_list (List.map mkVar x)))) ; New.observe_tac (fun _ _ -> str "prove_eq") (cont_tactic x) ])) let com_eqn uctx nb_arg eq_name functional_ref f_ref terminate_ref equation_lemma_type = let open CVars in let opacity = match terminate_ref with | GlobRef.ConstRef c -> is_opaque_constant c | _ -> anomaly ~label:"terminate_lemma" (Pp.str "not a constant.") in let evd = Evd.from_ctx uctx in let f_constr = constr_of_monomorphic_global f_ref in let equation_lemma_type = subst1 f_constr equation_lemma_type in let info = Declare.Info.make () in let cinfo = Declare.CInfo.make ~name:eq_name ~typ:(EConstr.of_constr equation_lemma_type) () in let lemma = Declare.Proof.start ~cinfo evd ~info in let lemma = fst @@ Declare.Proof.by (start_equation f_ref terminate_ref (fun x -> prove_eq (fun _ -> Proofview.tclUNIT ()) { nb_arg ; f_terminate = EConstr.of_constr (constr_of_monomorphic_global terminate_ref) ; f_constr = EConstr.of_constr f_constr ; concl_tac = Tacticals.New.tclIDTAC ; func = functional_ref ; info = instantiate_lambda Evd.empty (EConstr.of_constr (def_of_const (constr_of_monomorphic_global functional_ref))) (EConstr.of_constr f_constr :: List.map mkVar x) ; is_main_branch = true ; is_final = true ; values_and_bounds = [] ; eqs = [] ; forbidden_ids = [] ; acc_inv = lazy (assert false) ; acc_id = Id.of_string "____" ; args_assoc = [] ; f_id = Id.of_string "______" ; rec_arg_id = Id.of_string "______" ; is_mes = false ; ih = Id.of_string "______" })) lemma in let _ = Flags.silently (fun () -> let (_ : _ list) = Declare.Proof.save_regular ~proof:lemma ~opaque:opacity ~idopt:None in ()) () in () (* Pp.msgnl (fun _ _ -> str "eqn finished"); *) let recursive_definition ~interactive_proof ~is_mes function_name rec_impls type_of_f r rec_arg_num eq generate_induction_principle using_lemmas : Declare.Proof.t option = let open Term in let open Constr in let open CVars in let env = Global.env () in let evd = Evd.from_env env in let evd, function_type = interp_type_evars ~program_mode:false env evd type_of_f in let function_r = Sorts.Relevant in (* TODO relevance *) let env = EConstr.push_named (Context.Named.Declaration.LocalAssum (make_annot function_name function_r, function_type)) env in (* Pp.msgnl (str "function type := " ++ Printer.pr_lconstr function_type); *) let evd, ty = interp_type_evars ~program_mode:false env evd ~impls:rec_impls eq in let evd = Evd.minimize_universes evd in let equation_lemma_type = Reductionops.nf_betaiotazeta env evd (Evarutil.nf_evar evd ty) in let function_type = EConstr.to_constr ~abort_on_undefined_evars:false evd function_type in let equation_lemma_type = EConstr.Unsafe.to_constr equation_lemma_type in (* Pp.msgnl (fun _ _ -> str "lemma type := " ++ Printer.pr_lconstr equation_lemma_type ++ fnl ()); *) let res_vars, eq' = decompose_prod equation_lemma_type in let env_eq' = Environ.push_rel_context (List.map (fun (x, y) -> LocalAssum (x, y)) res_vars) env in let eq' = Reductionops.nf_zeta env_eq' evd (EConstr.of_constr eq') in let eq' = EConstr.Unsafe.to_constr eq' in let res = (* Pp.msgnl (fun _ _ -> str "res_var :=" ++ Printer.pr_lconstr_env (push_rel_context (List.map (function (x,t) -> (x,None,t)) res_vars) env) eq'); *) (* Pp.msgnl (fun _ _ -> str "rec_arg_num := " ++ str (fun _ _ -> string_of_int rec_arg_num)); *) (* Pp.msgnl (fun _ _ -> str "eq' := " ++ str (fun _ _ -> string_of_int rec_arg_num)); *) match Constr.kind eq' with | App (e, [|_; _; eq_fix|]) -> mkLambda ( make_annot (Name function_name) Sorts.Relevant , function_type , subst_var function_name (compose_lam res_vars eq_fix) ) | _ -> failwith "Recursive Definition (res not eq)" in let pre_rec_args, function_type_before_rec_arg = decompose_prod_n (rec_arg_num - 1) function_type in let _, rec_arg_type, _ = destProd function_type_before_rec_arg in let arg_types = List.rev_map snd (fst (decompose_prod_n (List.length res_vars) function_type)) in let equation_id = add_suffix function_name "_equation" in let functional_id = add_suffix function_name "_F" in let term_id = add_suffix function_name "_terminate" in let functional_ref = let univs = Evd.univ_entry ~poly:false evd in declare_fun functional_id Decls.(IsDefinition Definition) ~univs res in (* Refresh the global universes, now including those of _F *) let evd = Evd.from_env (Global.env ()) in let env_with_pre_rec_args = push_rel_context (List.map (function x, t -> LocalAssum (x, t)) pre_rec_args) env in let relation, evuctx = interp_constr env_with_pre_rec_args evd r in let evd = Evd.from_ctx evuctx in let tcc_lemma_name = add_suffix function_name "_tcc" in let tcc_lemma_constr = ref Undefined in (* let _ = Pp.msgnl (fun _ _ -> str "relation := " ++ Printer.pr_lconstr_env env_with_pre_rec_args relation) in *) let hook {Declare.Hook.S.uctx; _} = let term_ref = Nametab.locate (qualid_of_ident term_id) in let f_ref = declare_f function_name Decls.(IsProof Lemma) arg_types term_ref in let _ = Extraction_plugin.Table.extraction_inline true [qualid_of_ident term_id] in (* message "start second proof"; *) let stop = (* XXX: What is the correct way to get sign at hook time *) try com_eqn uctx (List.length res_vars) equation_id functional_ref f_ref term_ref (subst_var function_name equation_lemma_type); false with e when CErrors.noncritical e -> if do_observe () then Feedback.msg_debug (str "Cannot create equation Lemma " ++ CErrors.print e) else CErrors.user_err ~hdr:"Cannot create equation Lemma" ( str "Cannot create equation lemma." ++ spc () ++ str "This may be because the function is nested-recursive." ); true in if not stop then let eq_ref = Nametab.locate (qualid_of_ident equation_id) in let f_ref = destConst (constr_of_monomorphic_global f_ref) and functional_ref = destConst (constr_of_monomorphic_global functional_ref) and eq_ref = destConst (constr_of_monomorphic_global eq_ref) in generate_induction_principle f_ref tcc_lemma_constr functional_ref eq_ref rec_arg_num (EConstr.of_constr rec_arg_type) (nb_prod evd (EConstr.of_constr res)) relation in (* XXX STATE Why do we need this... why is the toplevel protection not enough *) funind_purify (fun () -> com_terminate interactive_proof tcc_lemma_name tcc_lemma_constr is_mes functional_ref (EConstr.of_constr rec_arg_type) relation rec_arg_num term_id using_lemmas (List.length res_vars) evd (Declare.Hook.make hook)) ()