(* Copyright (C) 2002, HELM Team. * * This file is part of HELM, an Hypertextual, Electronic * Library of Mathematics, developed at the Computer Science * Department, University of Bologna, Italy. * * HELM is free software; you can redistribute it and/or * modify it under the terms of the GNU General Public License * as published by the Free Software Foundation; either version 2 * of the License, or (at your option) any later version. * * HELM is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with HELM; if not, write to the Free Software * Foundation, Inc., 59 Temple Place - Suite 330, Boston, * MA 02111-1307, USA. * * For details, see the HELM World-Wide-Web page, * http://cs.unibo.it/helm/. *) (* $Id$ *) module C = Cic module U = UriManager module P = PrimitiveTactics module T = Tacticals module CR = CicReduction module PST = ProofEngineStructuralRules module PET = ProofEngineTypes module CTC = CicTypeChecker module CU = CicUniv module S = CicSubstitution let debug = false let debug_print = if debug then (fun x -> prerr_endline (Lazy.force x)) else (fun _ -> ()) ;; (* funzione generale di rilocazione dei riferimenti locali *) let relocate_term map t = let rec map_xnss k xnss = let imap (uri, t) = uri, map_term k t in List.map imap xnss and map_mss k mss = let imap = function | None -> None | Some t -> Some (map_term k t) in List.map imap mss and map_fs len k fs = let imap (name, i, ty, bo) = name, i, map_term k ty, map_term (k + len) bo in List.map imap fs and map_cfs len k cfs = let imap (name, ty, bo) = name, map_term k ty, map_term (k + len) bo in List.map imap cfs and map_term k = function | C.Rel m -> if m < k then C.Rel m else C.Rel (map (m - k)) | C.Sort _ as t -> t | C.Implicit _ as t -> t | C.Var (uri, xnss) -> C.Var (uri, map_xnss k xnss) | C.Const (uri, xnss) -> C.Const (uri, map_xnss k xnss) | C.MutInd (uri, tyno, xnss) -> C.MutInd (uri, tyno, map_xnss k xnss) | C.MutConstruct (uri, tyno, consno, xnss) -> C.MutConstruct (uri, tyno, consno, map_xnss k xnss) | C.Meta (i, mss) -> C.Meta(i, map_mss k mss) | C.Cast (te, ty) -> C.Cast (map_term k te, map_term k ty) | C.Appl ts -> C.Appl (List.map (map_term k) ts) | C.MutCase (sp, i, outty, t, pl) -> C.MutCase (sp, i, map_term k outty, map_term k t, List.map (map_term k) pl) | C.Prod (n, s, t) -> C.Prod (n, map_term k s, map_term (succ k) t) | C.Lambda (n, s, t) -> C.Lambda (n, map_term k s, map_term (succ k) t) | C.LetIn (n, s, t) -> C.LetIn (n, map_term k s, map_term (succ k) t) | C.Fix (i, fs) -> C.Fix (i, map_fs (List.length fs) k fs) | C.CoFix (i, cfs) -> C.CoFix (i, map_cfs (List.length cfs) k cfs) in map_term 0 t (* term ha tipo t1=t2; funziona solo se t1 e t2 hanno in testa costruttori diversi *) let discriminate_tac ~term = let true_URI = match LibraryObjects.true_URI () with Some uri -> uri | None -> raise (PET.Fail (lazy "You need to register the default \"true\" definition first. Please use the \"default\" command")) in let false_URI = match LibraryObjects.false_URI () with Some uri -> uri | None -> raise (PET.Fail (lazy "You need to register the default \"false\" definition first. Please use the \"default\" command")) in let fail msg = raise (PET.Fail (lazy ("Discriminate: " ^ msg))) in let find_discriminating_consno t1 t2 = let rec aux t1 t2 = match t1, t2 with | C.MutConstruct _, C.MutConstruct _ when t1 = t2 -> None | C.Appl ((C.MutConstruct _ as constr1) :: args1), C.Appl ((C.MutConstruct _ as constr2) :: args2) when constr1 = constr2 -> let rec aux_list l1 l2 = match l1, l2 with | [], [] -> None | hd1 :: tl1, hd2 :: tl2 -> (match aux hd1 hd2 with | None -> aux_list tl1 tl2 | Some _ as res -> res) | _ -> (* same constructor applied to a different number of args *) assert false in aux_list args1 args2 | ((C.MutConstruct (_,_,consno1,subst1)), (C.MutConstruct (_,_,consno2,subst2))) | ((C.MutConstruct (_,_,consno1,subst1)), (C.Appl ((C.MutConstruct (_,_,consno2,subst2)) :: _))) | ((C.Appl ((C.MutConstruct (_,_,consno1,subst1)) :: _)), (C.MutConstruct (_,_,consno2,subst2))) | ((C.Appl ((C.MutConstruct (_,_,consno1,subst1)) :: _)), (C.Appl ((C.MutConstruct (_,_,consno2,subst2)) :: _))) when (consno1 <> consno2) || (subst1 <> subst2) -> Some consno2 | _ -> fail "not a discriminable equality" in aux t1 t2 in let mk_branches_and_outtype turi typeno consno context args = (* a list of "True" except for the element in position consno which * is "False" *) match fst (CicEnvironment.get_obj CicUniv.empty_ugraph turi) with | C.InductiveDefinition (ind_type_list,_,paramsno,_) -> let _,_,rty,constructor_list = List.nth ind_type_list typeno in let false_constr_id,_ = List.nth constructor_list (consno - 1) in let branches = List.map (fun (id,cty) -> (* dubbio: e' corretto ridurre in questo context ??? *) let red_ty = CR.whd context cty in let rec aux t k = match t with | C.Prod (_,_,target) when (k <= paramsno) -> S.subst (List.nth args (k-1)) (aux target (k+1)) | C.Prod (binder,source,target) when (k > paramsno) -> C.Lambda (binder, source, (aux target (k+1))) | _ -> if (id = false_constr_id) then (C.MutInd(false_URI,0,[])) else (C.MutInd(true_URI,0,[])) in (S.lift 1 (aux red_ty 1))) constructor_list in let outtype = let seed = ref 0 in let rec mk_lambdas rev_left_args = function 0, args, C.Prod (_,so,ta) -> C.Lambda (C.Name (incr seed; "x" ^ string_of_int !seed), so, mk_lambdas rev_left_args (0,args,ta)) | 0, args, C.Sort _ -> let rec mk_rels = function 0 -> [] | n -> C.Rel n :: mk_rels (n - 1) in let argsno = List.length args in C.Lambda (C.Name "x", (if argsno + List.length rev_left_args > 0 then C.Appl (C.MutInd (turi, typeno, []) :: (List.map (S.lift (argsno + 1)) (List.rev rev_left_args)) @ mk_rels argsno) else C.MutInd (turi,typeno,[])), C.Sort C.Prop) | 0, _, _ -> assert false (* seriously screwed up *) | n, he::tl, C.Prod (_,_,ta) -> mk_lambdas (he::rev_left_args)(n-1,tl,S.subst he ta) | n,_,_ -> assert false (* we should probably reduce in some context *) in mk_lambdas [] (paramsno, args, rty) in branches, outtype | _ -> assert false in let discriminate'_tac ~term status = let (proof, goal) = status in let _,metasenv,_subst,_,_, _ = proof in let _,context,_ = CicUtil.lookup_meta goal metasenv in let termty,_ = CTC.type_of_aux' metasenv context term CicUniv.empty_ugraph in match termty with | C.Appl [(C.MutInd (equri, 0, [])) ; tty ; t1 ; t2] when LibraryObjects.is_eq_URI equri -> let turi,typeno,exp_named_subst,args = match tty with | (C.MutInd (turi,typeno,exp_named_subst)) -> turi,typeno,exp_named_subst,[] | (C.Appl (C.MutInd (turi,typeno,exp_named_subst)::args)) -> turi,typeno,exp_named_subst,args | _ -> fail "not a discriminable equality" in let consno = match find_discriminating_consno t1 t2 with | Some consno -> consno | None -> fail "discriminating terms are structurally equal" in let branches,outtype = mk_branches_and_outtype turi typeno consno context args in PET.apply_tactic (T.then_ ~start:(EliminationTactics.elim_type_tac (C.MutInd (false_URI, 0, []))) ~continuation: (T.then_ ~start: (ReductionTactics.change_tac ~pattern:(PET.conclusion_pattern None) (fun _ m u -> C.Appl [ C.Lambda ( C.Name "x", tty, C.MutCase (turi, typeno, outtype, (C.Rel 1), branches)); t2 ], m, u)) ~continuation: (T.then_ ~start: (EqualityTactics.rewrite_simpl_tac ~direction:`RightToLeft ~pattern:(PET.conclusion_pattern None) term []) ~continuation: (IntroductionTactics.constructor_tac ~n:1)))) status | _ -> fail "not an equality" in PET.mk_tactic (discriminate'_tac ~term) let exn_nonproj = PET.Fail (lazy "Injection: not a projectable equality") let exn_noneq = PET.Fail (lazy "Injection: not an equality") let exn_nothingtodo = PET.Fail (lazy "Nothing to do") let exn_discrnonind = PET.Fail (lazy "Discriminate: object is not an Inductive Definition: it's imposible") let exn_injwronggoal = PET.Fail (lazy "Injection: goal after cut is not correct") let exn_noneqind = PET.Fail (lazy "Injection: not an equality over elements of an inductive type") let pp ctx t = let names = List.map (function Some (n,_) -> Some n | None -> None) ctx in CicPp.pp t names let rec injection_tac ~term ~i ~liftno ~continuation = let give_name seed = function | C.Name _ as name -> name | C.Anonymous -> C.Name (incr seed; "y" ^ string_of_int !seed) in let rec mk_rels = function | 0 -> [] | n -> C.Rel n :: (mk_rels (n - 1)) in let injection_tac ~term ~i status = let (proof, goal) = status in (* precondizione: t1 e t2 hanno in testa lo stesso costruttore ma * differiscono (o potrebbero differire?) nell'i-esimo parametro * del costruttore *) let term = S.lift liftno term in let _,metasenv,_subst,_,_, _ = proof in let _,context,_ = CicUtil.lookup_meta goal metasenv in let termty,_ = CTC.type_of_aux' metasenv context term CicUniv.empty_ugraph in debug_print (lazy ("\ninjection1 su : " ^ pp context termty)); match termty with (* an equality *) | C.Appl [(C.MutInd (equri, 0, [])) ; tty ; t1 ; t2] when LibraryObjects.is_eq_URI equri -> let turi,typeno,ens,params = match tty with (* some inductive type *) | C.MutInd (turi,typeno,ens) -> turi,typeno,ens,[] | C.Appl (C.MutInd (turi,typeno,ens)::params) -> turi,typeno,ens,params | _ -> raise exn_noneqind in let t1',t2',consno = (* sono i due sottotermini che differiscono *) match t1,t2 with | C.Appl ((C.MutConstruct (uri1,typeno1,consno1,ens1))::applist1), C.Appl ((C.MutConstruct (uri2,typeno2,consno2,ens2))::applist2) when (uri1 = uri2) && (typeno1 = typeno2) && (consno1 = consno2) && (ens1 = ens2) -> (* controllo ridondante *) List.nth applist1 (i-1),List.nth applist2 (i-1),consno2 | _ -> assert false in let tty',_ = CTC.type_of_aux' metasenv context t1' CU.empty_ugraph in let patterns,outtype = match fst (CicEnvironment.get_obj CicUniv.empty_ugraph turi) with | C.InductiveDefinition (ind_type_list,_,paramsno,_)-> let left_params, right_params = HExtlib.split_nth paramsno params in let _,_,_,constructor_list = List.nth ind_type_list typeno in let i_constr_id,_ = List.nth constructor_list (consno - 1) in let patterns = let seed = ref 0 in List.map (function (id,cty) -> let reduced_cty = CR.whd context cty in let rec aux k = function | C.Prod (_,_,tgt) when k <= paramsno -> let left = List.nth left_params (k-1) in aux (k+1) (S.subst left tgt) | C.Prod (binder,source,target) when k > paramsno -> let binder' = give_name seed binder in C.Lambda (binder',source,(aux (k+1) target)) | _ -> let nr_param_constr = k - paramsno - 1 in if id = i_constr_id then C.Rel (k - i) else S.lift nr_param_constr t1' (* + 1 per liftare anche il lambda aggiunto * esternamente al case *) in S.lift 1 (aux 1 reduced_cty)) constructor_list in (* this code should be taken from cases_tac *) let outtype = let seed = ref 0 in let rec to_lambdas te head = match CR.whd context te with | C.Prod (binder,so,ta) -> let binder' = give_name seed binder in C.Lambda (binder',so,to_lambdas ta head) | _ -> head in let rec skip_prods params te = match params, CR.whd context te with | [], _ -> te | left::tl, C.Prod (_,_,ta) -> skip_prods tl (S.subst left ta) | _, _ -> assert false in let abstracted_tty = let tty = List.fold_left (fun x y -> S.subst y x) tty left_params in (* non lift, ma subst coi left! *) match S.lift 1 tty with | C.MutInd _ as tty' -> tty' | C.Appl l -> let keep,abstract = HExtlib.split_nth (paramsno +1) l in let keep = List.map (S.lift paramsno) keep in C.Appl (keep@mk_rels (List.length abstract)) | _ -> assert false in match ind_type_list with | [] -> assert false | (_,_,ty,_)::_ -> (* this is in general wrong, do as in cases_tac *) to_lambdas (skip_prods left_params ty) (C.Lambda (C.Name "cased", abstracted_tty, (* here we should capture right parameters *) (* 1 for his Lambda, one for the Lambda outside the match * and then one for each to_lambda *) S.lift (2+List.length right_params) tty')) in patterns,outtype | _ -> raise exn_discrnonind in let cutted = C.Appl [C.MutInd (equri,0,[]) ; tty' ; t1' ; t2'] in let changed = C.Appl [ C.Lambda (C.Name "x", tty, C.MutCase (turi,typeno,outtype,C.Rel 1,patterns)) ; t1] in (* check if cutted and changed are well typed and if t1' ~ changed *) let go_on = try let _,g = CTC.type_of_aux' metasenv context cutted CicUniv.empty_ugraph in let _,g = CTC.type_of_aux' metasenv context changed g in fst (CR.are_convertible ~metasenv context t1' changed g) with | CTC.TypeCheckerFailure _ -> false in if not go_on then PET.apply_tactic T.id_tac status else (debug_print (lazy ("CUT: " ^ pp context cutted)); PET.apply_tactic (T.thens ~start: (P.cut_tac cutted) ~continuations: [injection1_tac ~first_time:false ~liftno:0 ~term:(C.Rel 1) ~continuation: (fun ~liftno:x -> continuation ~liftno:(liftno+1+x)) (* here I need to lift all the continuations by 1; since I am setting back liftno to 0, I actually need to lift all the continuations by liftno + 1 *) ;T.then_ ~start:(PET.mk_tactic (fun status -> debug_print (lazy "riempo il cut"); let (proof, goal) = status in let _,metasenv,_subst,_,_, _ = proof in let _,context,gty =CicUtil.lookup_meta goal metasenv in let gty = Unshare.unshare gty in let new_t1' = match gty with | (C.Appl (C.MutInd (_,_,_)::_::t::_)) -> t | _ -> raise exn_injwronggoal in debug_print (lazy ("metto questo: " ^ pp context changed)); debug_print (lazy ("al posto di questo: " ^ pp context new_t1')); debug_print (lazy ("nel goal: " ^ pp context gty)); debug_print (lazy ("nel contesto:\n" ^ CicPp.ppcontext context)); debug_print (lazy ("e poi rewrite con: "^pp context term)); let rc = PET.apply_tactic (ReductionTactics.change_tac ~pattern:(None, [], Some (ProofEngineHelpers.pattern_of ~term:gty [new_t1'])) (fun _ m u -> changed,m,u)) status in rc )) ~continuation: (T.then_ ~start: (EqualityTactics.rewrite_simpl_tac ~direction:`LeftToRight ~pattern:(PET.conclusion_pattern None) term []) ~continuation:EqualityTactics.reflexivity_tac) ]) status) | _ -> raise exn_noneq in PET.mk_tactic (injection_tac ~term ~i) and injection1_tac ~first_time ~term ~liftno ~continuation = let are_convertible hd1 hd2 metasenv context = fst (CR.are_convertible ~metasenv context hd1 hd2 CicUniv.empty_ugraph) in let injection1_tac ~term status = let (proof, goal) = status in let _,metasenv,_subst, _,_, _ = proof in let _,context,_ = CicUtil.lookup_meta goal metasenv in let term = S.lift liftno term in let termty,_ = CTC.type_of_aux' metasenv context term CicUniv.empty_ugraph in debug_print (lazy ("\ninjection su: " ^ pp context termty)); let tac = match termty with | C.Appl [(C.MutInd (equri, 0, [])) ; tty ; t1 ; t2] when LibraryObjects.is_eq_URI equri -> begin match (CR.whd ~delta:true context tty) with | C.MutInd (turi,typeno,ens) | C.Appl (C.MutInd (turi,typeno,ens)::_) -> begin match t1,t2 with | C.MutConstruct (uri1,typeno1,consno1,ens1), C.MutConstruct (uri2,typeno2,consno2,ens2) when (uri1 = uri2) && (typeno1 = typeno2) && (consno1 = consno2) && (ens1 = ens2) -> if first_time then raise exn_nothingtodo else continuation ~liftno | C.Appl ((C.MutConstruct (uri1,typeno1,consno1,ens1))::applist1), C.Appl ((C.MutConstruct (uri2,typeno2,consno2,ens2))::applist2) when (uri1 = uri2) && (typeno1 = typeno2) && (consno1 = consno2) && (ens1 = ens2) -> let rec traverse_list i l1 l2 = match l1,l2 with | [],[] when first_time -> continuation | [],[] -> begin match term with | C.Rel n -> begin match List.nth context (n-1) with | Some (C.Name id,_) -> fun ~liftno -> T.then_ ~start:(PST.clear ~hyps:[id]) ~continuation:(continuation ~liftno) | _ -> assert false end | _ -> assert false end | hd1::tl1,hd2::tl2 -> if are_convertible hd1 hd2 metasenv context then traverse_list (i+1) tl1 tl2 else injection_tac ~i ~term ~continuation:(traverse_list (i+1) tl1 tl2) | _ -> assert false (* i 2 termini hanno in testa lo stesso costruttore, * ma applicato a un numero diverso di termini *) in traverse_list 1 applist1 applist2 ~liftno | C.MutConstruct (uri1,typeno1,consno1,ens1), C.MutConstruct (uri2,typeno2,consno2,ens2) | C.MutConstruct (uri1,typeno1,consno1,ens1), C.Appl ((C.MutConstruct (uri2,typeno2,consno2,ens2))::_) | C.Appl ((C.MutConstruct (uri1,typeno1,consno1,ens1))::_), C.MutConstruct (uri2,typeno2,consno2,ens2) | C.Appl ((C.MutConstruct (uri1,typeno1,consno1,ens1))::_), C.Appl ((C.MutConstruct (uri2,typeno2,consno2,ens2))::_) when (consno1 <> consno2) || (ens1 <> ens2) -> discriminate_tac ~term | _ when not first_time -> continuation ~liftno | _ (* when first_time *) -> match term with | Cic.Rel i -> let name = match List.nth context (i-1) with | Some (Cic.Name s, Cic.Def _) -> s | Some (Cic.Name s, Cic.Decl _) -> s | _ -> assert false in T.then_ ~start:(ReductionTactics.simpl_tac ~pattern:(None,[name,Cic.Implicit (Some `Hole)],None)) ~continuation:(injection1_tac ~first_time:false ~term ~liftno ~continuation) | _ -> raise exn_nonproj end | _ when not first_time -> continuation ~liftno | _ (* when first_time *) -> raise exn_nonproj end | _ -> raise exn_nonproj in PET.apply_tactic tac status in PET.mk_tactic (injection1_tac ~term) (* destruct performs either injection or discriminate *) (* equivalent to Coq's "analyze equality" *) let destruct_tac = injection1_tac ~first_time:true ~liftno:0 ~continuation:(fun ~liftno -> T.id_tac)