(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* count (n+1) t | LetIn(_,a,_,t) -> count n (subst1 a t) | Cast(c,_,_) -> count n c | _ -> n in count 0 x (*********************************************) (* Tactics *) (*********************************************) (****************************************) (* General functions *) (****************************************) (* let get_pairs_from_bindings = let pair_from_binding = function | [(Bindings binds)] -> binds | _ -> error "not a binding list!" in List.map pair_from_binding *) let string_of_inductive c = try match kind_of_term c with | Ind ind_sp -> let (mib,mip) = Global.lookup_inductive ind_sp in string_of_id mip.mind_typename | _ -> raise Bound with Bound -> error "Bound head variable" let rec head_constr_bound t l = let t = strip_outer_cast(collapse_appl t) in match kind_of_term t with | Prod (_,_,c2) -> head_constr_bound c2 l | LetIn (_,_,_,c2) -> head_constr_bound c2 l | App (f,args) -> head_constr_bound f (Array.fold_right (fun a l -> a::l) args l) | Const _ -> t::l | Ind _ -> t::l | Construct _ -> t::l | Var _ -> t::l | _ -> raise Bound let head_constr c = try head_constr_bound c [] with Bound -> error "Bound head variable" (* let bad_tactic_args s l = raise (RefinerError (BadTacticArgs (s,l))) *) (******************************************) (* Primitive tactics *) (******************************************) let introduction = Tacmach.introduction let intro_replacing = Tacmach.intro_replacing let internal_cut = Tacmach.internal_cut let internal_cut_rev = Tacmach.internal_cut_rev let refine = Tacmach.refine let convert_concl = Tacmach.convert_concl let convert_hyp = Tacmach.convert_hyp let thin = Tacmach.thin let thin_body = Tacmach.thin_body (* Moving hypotheses *) let move_hyp = Tacmach.move_hyp (* Renaming hypotheses *) let rename_hyp = Tacmach.rename_hyp (* Refine as a fixpoint *) let mutual_fix = Tacmach.mutual_fix let fix ido n = match ido with | None -> mutual_fix (Pfedit.get_current_proof_name ()) n [] | Some id -> mutual_fix id n [] (* Refine as a cofixpoint *) let mutual_cofix = Tacmach.mutual_cofix let cofix = function | None -> mutual_cofix (Pfedit.get_current_proof_name ()) [] | Some id -> mutual_cofix id [] (**************************************************************) (* Reduction and conversion tactics *) (**************************************************************) type tactic_reduction = env -> evar_map -> constr -> constr (* The following two tactics apply an arbitrary reduction function either to the conclusion or to a certain hypothesis *) let reduct_in_concl (redfun,sty) gl = convert_concl_no_check (pf_reduce redfun gl (pf_concl gl)) sty gl let reduct_in_hyp redfun (id,_,(where,where')) gl = let (_,c, ty) = pf_get_hyp gl id in let redfun' = (*under_casts*) (pf_reduce redfun gl) in match c with | None -> if where = InHypValueOnly then errorlabstrm "" (pr_id id ++ str "has no value"); if Options.do_translate () then where' := Some where; convert_hyp_no_check (id,None,redfun' ty) gl | Some b -> let where = if !Options.v7 & where = InHyp then InHypValueOnly else where in let b' = if where <> InHypTypeOnly then redfun' b else b in let ty' = if where <> InHypValueOnly then redfun' ty else ty in if Options.do_translate () then where' := Some where; convert_hyp_no_check (id,Some b',ty') gl let reduct_option redfun = function | Some id -> reduct_in_hyp (fst redfun) id | None -> reduct_in_concl redfun (* The following tactic determines whether the reduction function has to be applied to the conclusion or to the hypotheses. *) let redin_combinator redfun = onClauses (reduct_option redfun) (* Now we introduce different instances of the previous tacticals *) let change_and_check cv_pb t env sigma c = if is_fconv cv_pb env sigma t c then t else errorlabstrm "convert-check-hyp" (str "Not convertible") (* Use cumulutavity only if changing the conclusion not a subterm *) let change_on_subterm cv_pb t = function | None -> change_and_check cv_pb t | Some occl -> contextually false occl (change_and_check Reduction.CONV t) let change_in_concl occl t = reduct_in_concl ((change_on_subterm Reduction.CUMUL t occl),DEFAULTcast) let change_in_hyp occl t = reduct_in_hyp (change_on_subterm Reduction.CONV t occl) let change_option occl t = function Some id -> change_in_hyp occl t id | None -> change_in_concl occl t let change occl c cls = (match cls, occl with ({onhyps=(Some(_::_::_)|None)}|{onhyps=Some(_::_);onconcl=true}), Some _ -> error "No occurrences expected when changing several hypotheses" | _ -> ()); onClauses (change_option occl c) cls (* Pour usage interne (le niveau User est pris en compte par reduce) *) let red_in_concl = reduct_in_concl (red_product,DEFAULTcast) let red_in_hyp = reduct_in_hyp red_product let red_option = reduct_option (red_product,DEFAULTcast) let hnf_in_concl = reduct_in_concl (hnf_constr,DEFAULTcast) let hnf_in_hyp = reduct_in_hyp hnf_constr let hnf_option = reduct_option (hnf_constr,DEFAULTcast) let simpl_in_concl = reduct_in_concl (nf,DEFAULTcast) let simpl_in_hyp = reduct_in_hyp nf let simpl_option = reduct_option (nf,DEFAULTcast) let normalise_in_concl = reduct_in_concl (compute,DEFAULTcast) let normalise_in_hyp = reduct_in_hyp compute let normalise_option = reduct_option (compute,DEFAULTcast) let normalise_vm_in_concl = reduct_in_concl (Redexpr.cbv_vm,VMcast) let unfold_in_concl loccname = reduct_in_concl (unfoldn loccname,DEFAULTcast) let unfold_in_hyp loccname = reduct_in_hyp (unfoldn loccname) let unfold_option loccname = reduct_option (unfoldn loccname,DEFAULTcast) let pattern_option l = reduct_option (pattern_occs l,DEFAULTcast) (* A function which reduces accordingly to a reduction expression, as the command Eval does. *) let needs_check = function (* Expansion is not necessarily well-typed: e.g. expansion of t into x is not well-typed in [H:(P t); x:=t |- G] because x is defined after H *) | Fold _ -> true | _ -> false let reduce redexp cl goal = (if needs_check redexp then with_check else (fun x -> x)) (redin_combinator (Redexpr.reduction_of_red_expr redexp) cl) goal (* Unfolding occurrences of a constant *) let unfold_constr = function | ConstRef sp -> unfold_in_concl [[],EvalConstRef sp] | VarRef id -> unfold_in_concl [[],EvalVarRef id] | _ -> errorlabstrm "unfold_constr" (str "Cannot unfold a non-constant.") (*******************************************) (* Introduction tactics *) (*******************************************) let fresh_id avoid id gl = next_global_ident_away true id (avoid@(pf_ids_of_hyps gl)) let id_of_name_with_default s = function | Anonymous -> id_of_string s | Name id -> id let default_id gl = function | (name,None,t) -> (match kind_of_term (pf_whd_betadeltaiota gl (pf_type_of gl t)) with | Sort (Prop _) -> (id_of_name_with_default "H" name) | Sort (Type _) -> (id_of_name_with_default "X" name) | _ -> anomaly "Wrong sort") | (name,Some b,_) -> id_of_name_using_hdchar (pf_env gl) b name (* Non primitive introduction tactics are treated by central_intro There is possibly renaming, with possibly names to avoid and possibly a move to do after the introduction *) type intro_name_flag = | IntroAvoid of identifier list | IntroBasedOn of identifier * identifier list | IntroMustBe of identifier let find_name decl gl = function | IntroAvoid idl -> let id = fresh_id idl (default_id gl decl) gl in id | IntroBasedOn (id,idl) -> fresh_id idl id gl | IntroMustBe id -> let id' = fresh_id [] id gl in if id' <> id then error ((string_of_id id)^" is already used"); id' let build_intro_tac id = function | None -> introduction id | Some dest -> tclTHEN (introduction id) (move_hyp true id dest) let rec intro_gen name_flag move_flag force_flag gl = match kind_of_term (pf_concl gl) with | Prod (name,t,_) -> build_intro_tac (find_name (name,None,t) gl name_flag) move_flag gl | LetIn (name,b,t,_) -> build_intro_tac (find_name (name,Some b,t) gl name_flag) move_flag gl | _ -> if not force_flag then raise (RefinerError IntroNeedsProduct); try tclTHEN (reduce (Red true) onConcl) (intro_gen name_flag move_flag force_flag) gl with Redelimination -> errorlabstrm "Intro" (str "No product even after head-reduction") let intro_mustbe_force id = intro_gen (IntroMustBe id) None true let intro_using id = intro_gen (IntroBasedOn (id,[])) None false let intro_force force_flag = intro_gen (IntroAvoid []) None force_flag let intro = intro_force false let introf = intro_force true let introf_move_name destopt = intro_gen (IntroAvoid []) destopt true (* For backwards compatibility *) let central_intro = intro_gen (**** Multiple introduction tactics ****) let rec intros_using = function [] -> tclIDTAC | str::l -> tclTHEN (intro_using str) (intros_using l) let intros = tclREPEAT (intro_force false) let intro_erasing id = tclTHEN (thin [id]) (introduction id) let intros_replacing ids gls = let rec introrec = function | [] -> tclIDTAC | id::tl -> (tclTHEN (tclORELSE (intro_replacing id) (tclORELSE (intro_erasing id) (* ?? *) (intro_using id))) (introrec tl)) in introrec ids gls (* User-level introduction tactics *) let intro_move idopt idopt' = match idopt with | None -> intro_gen (IntroAvoid []) idopt' true | Some id -> intro_gen (IntroMustBe id) idopt' true let pf_lookup_hypothesis_as_renamed env ccl = function | AnonHyp n -> pf_lookup_index_as_renamed env ccl n | NamedHyp id -> pf_lookup_name_as_renamed env ccl id let pf_lookup_hypothesis_as_renamed_gen red h gl = let env = pf_env gl in let rec aux ccl = match pf_lookup_hypothesis_as_renamed env ccl h with | None when red -> aux ((fst (Redexpr.reduction_of_red_expr (Red true))) env (project gl) ccl) | x -> x in try aux (pf_concl gl) with Redelimination -> None let is_quantified_hypothesis id g = match pf_lookup_hypothesis_as_renamed_gen true (NamedHyp id) g with | Some _ -> true | None -> false let msg_quantified_hypothesis = function | NamedHyp id -> str "hypothesis " ++ pr_id id | AnonHyp n -> int n ++ str (match n with 1 -> "st" | 2 -> "nd" | _ -> "th") ++ str " non dependent hypothesis" let depth_of_quantified_hypothesis red h gl = match pf_lookup_hypothesis_as_renamed_gen red h gl with | Some depth -> depth | None -> errorlabstrm "lookup_quantified_hypothesis" (str "No " ++ msg_quantified_hypothesis h ++ str " in current goal" ++ if red then str " even after head-reduction" else mt ()) let intros_until_gen red h g = tclDO (depth_of_quantified_hypothesis red h g) intro g let intros_until_id id = intros_until_gen true (NamedHyp id) let intros_until_n_gen red n = intros_until_gen red (AnonHyp n) let intros_until = intros_until_gen true let intros_until_n = intros_until_n_gen true let intros_until_n_wored = intros_until_n_gen false let try_intros_until tac = function | NamedHyp id -> tclTHEN (tclTRY (intros_until_id id)) (tac id) | AnonHyp n -> tclTHEN (intros_until_n n) (onLastHyp tac) let rec intros_move = function | [] -> tclIDTAC | (hyp,destopt) :: rest -> tclTHEN (intro_gen (IntroMustBe hyp) destopt false) (intros_move rest) let dependent_in_decl a (_,c,t) = match c with | None -> dependent a t | Some body -> dependent a body || dependent a t let move_to_rhyp rhyp gl = let rec get_lhyp lastfixed depdecls = function | [] -> (match rhyp with | None -> lastfixed | Some h -> anomaly ("Hypothesis should occur: "^ (string_of_id h))) | (hyp,c,typ) as ht :: rest -> if Some hyp = rhyp then lastfixed else if List.exists (occur_var_in_decl (pf_env gl) hyp) depdecls then get_lhyp lastfixed (ht::depdecls) rest else get_lhyp (Some hyp) depdecls rest in let sign = pf_hyps gl in let (hyp,c,typ as decl) = List.hd sign in match get_lhyp None [decl] (List.tl sign) with | None -> tclIDTAC gl | Some hypto -> move_hyp true hyp hypto gl let rec intros_rmove = function | [] -> tclIDTAC | (hyp,destopt) :: rest -> tclTHENLIST [ introduction hyp; move_to_rhyp destopt; intros_rmove rest ] (****************************************************) (* Resolution tactics *) (****************************************************) (* Refinement tactic: unification with the head of the head normal form * of the type of a term. *) let apply_type hdcty argl gl = refine (applist (mkCast (Evarutil.mk_new_meta(),DEFAULTcast, hdcty),argl)) gl let apply_term hdc argl gl = refine (applist (hdc,argl)) gl let bring_hyps hyps = if hyps = [] then Refiner.tclIDTAC else (fun gl -> let newcl = List.fold_right mkNamedProd_or_LetIn hyps (pf_concl gl) in let f = mkCast (Evarutil.mk_new_meta(),DEFAULTcast, newcl) in refine_no_check (mkApp (f, instance_from_named_context hyps)) gl) (* Resolution with missing arguments *) let apply_with_bindings (c,lbind) gl = (* The actual type of the theorem. It will be matched against the goal. If this fails, then the head constant will be unfolded step by step. *) let thm_ty0 = nf_betaiota (pf_type_of gl c) in let rec try_apply thm_ty = try let n = nb_prod thm_ty - nb_prod (pf_concl gl) in if n<0 then error "Apply: theorem has not enough premisses."; let clause = make_clenv_binding_apply gl n (c,thm_ty) lbind in Clenvtac.res_pf clause gl with (Pretype_errors.PretypeError _|RefinerError _|UserError _|Failure _) as exn -> let red_thm = try red_product (pf_env gl) (project gl) thm_ty with (Redelimination | UserError _) -> raise exn in try_apply red_thm in try try_apply thm_ty0 with (Pretype_errors.PretypeError _|RefinerError _|UserError _|Failure _) -> (* Last chance: if the head is a variable, apply may try second order unification *) let clause = make_clenv_binding_apply gl (-1) (c,thm_ty0) lbind in Clenvtac.res_pf clause gl let apply c = apply_with_bindings (c,NoBindings) let apply_list = function | c::l -> apply_with_bindings (c,ImplicitBindings l) | _ -> assert false (* Resolution with no reduction on the type *) let apply_without_reduce c gl = let clause = mk_clenv_type_of gl c in res_pf clause gl (* A useful resolution tactic which, if c:A->B, transforms |- C into |- B -> C and |- A ------------------- Gamma |- c : A -> B Gamma |- ?2 : A ---------------------------------------- Gamma |- B Gamma |- ?1 : B -> C ----------------------------------------------------- Gamma |- ? : C Ltac lapply c := let ty := check c in match eval hnf in ty with ?A -> ?B => cut B; [ idtac | apply c ] end. *) let cut_and_apply c gl = let goal_constr = pf_concl gl in match kind_of_term (pf_hnf_constr gl (pf_type_of gl c)) with | Prod (_,c1,c2) when not (dependent (mkRel 1) c2) -> tclTHENLAST (apply_type (mkProd (Anonymous,c2,goal_constr)) [mkMeta(new_meta())]) (apply_term c [mkMeta (new_meta())]) gl | _ -> error "Imp_elim needs a non-dependent product" (**************************) (* Cut tactics *) (**************************) let assert_tac first na c gl = match kind_of_term (hnf_type_of gl c) with | Sort s -> let id = match na with | Anonymous -> let d = match s with Prop _ -> "H" | Type _ -> "X" in fresh_id [] (id_of_string d) gl | Name id -> id in (if first then internal_cut else internal_cut_rev) id c gl | _ -> error "Not a proposition or a type" let true_cut = assert_tac true let cut c gl = match kind_of_term (hnf_type_of gl c) with | Sort _ -> let id=next_name_away_with_default "H" Anonymous (pf_ids_of_hyps gl) in let t = mkProd (Anonymous, c, pf_concl gl) in tclTHENFIRST (internal_cut_rev id c) (tclTHEN (apply_type t [mkVar id]) (thin [id])) gl | _ -> error "Not a proposition or a type" let cut_intro t = tclTHENFIRST (cut t) intro let cut_replacing id t tac = tclTHENS (cut t) [tclORELSE (intro_replacing id) (tclORELSE (intro_erasing id) (intro_using id)); tac (refine_no_check (mkVar id)) ] let cut_in_parallel l = let rec prec = function | [] -> tclIDTAC | h::t -> tclTHENFIRST (cut h) (prec t) in prec (List.rev l) (**************************) (* Generalize tactics *) (**************************) let generalize_goal gl c cl = let t = pf_type_of gl c in match kind_of_term c with | Var id -> (* The choice of remembering or not a non dependent name has an impact on the future Intro naming strategy! *) (* if dependent c cl then mkNamedProd id t cl else mkProd (Anonymous,t,cl) *) mkNamedProd id t cl | _ -> let cl' = subst_term c cl in if noccurn 1 cl' then mkProd (Anonymous,t,cl) (* On ne se casse pas la tete : on prend pour nom de variable la premiere lettre du type, meme si "ci" est une constante et qu'on pourrait prendre directement son nom *) else prod_name (Global.env()) (Anonymous, t, cl') let generalize_dep c gl = let env = pf_env gl in let sign = pf_hyps gl in let init_ids = ids_of_named_context (Global.named_context()) in let rec seek d toquant = if List.exists (fun (id,_,_) -> occur_var_in_decl env id d) toquant or dependent_in_decl c d then d::toquant else toquant in let to_quantify = Sign.fold_named_context seek sign ~init:[] in let to_quantify_rev = List.rev to_quantify in let qhyps = List.map (fun (id,_,_) -> id) to_quantify_rev in let tothin = List.filter (fun id -> not (List.mem id init_ids)) qhyps in let tothin' = match kind_of_term c with | Var id when mem_named_context id sign & not (List.mem id init_ids) -> id::tothin | _ -> tothin in let cl' = it_mkNamedProd_or_LetIn (pf_concl gl) to_quantify in let cl'' = generalize_goal gl c cl' in let args = Array.to_list (instance_from_named_context to_quantify_rev) in tclTHEN (apply_type cl'' (c::args)) (thin (List.rev tothin')) gl let generalize lconstr gl = let newcl = List.fold_right (generalize_goal gl) lconstr (pf_concl gl) in apply_type newcl lconstr gl (* Faudra-t-il une version avec plusieurs args de generalize_dep ? Cela peut-être troublant de faire "Generalize Dependent H n" dans "n:nat; H:n=n |- P(n)" et d'échouer parce que H a disparu après la généralisation dépendante par n. let quantify lconstr = List.fold_right (fun com tac -> tclTHEN tac (tactic_com generalize_dep c)) lconstr tclIDTAC *) (* A dependent cut rule à la sequent calculus ------------------------------------------ Sera simplifiable le jour où il y aura un let in primitif dans constr [letin_tac b na c (occ_hyp,occ_ccl) gl] transforms [...x1:T1(c),...,x2:T2(c),... |- G(c)] into [...x:T;x1:T1(x),...,x2:T2(x),... |- G(x)] if [b] is false or [...x:=c:T;x1:T1(x),...,x2:T2(x),... |- G(x)] if [b] is true [occ_hyp,occ_ccl] tells which occurrences of [c] have to be substituted; if [occ_hyp = []] and [occ_ccl = None] then [c] is substituted wherever it occurs, otherwise [c] is substituted only in hyps present in [occ_hyps] at the specified occurrences (everywhere if the list of occurrences is empty), and in the goal at the specified occurrences if [occ_goal] is not [None]; if name = Anonymous, the name is build from the first letter of the type; The tactic first quantify the goal over x1, x2,... then substitute then re-intro x1, x2,... at their initial place ([marks] is internally used to remember the place of x1, x2, ...: it is the list of hypotheses on the left of each x1, ...). *) let occurrences_of_hyp id cls = let rec hyp_occ = function [] -> None | (id',occs,hl)::_ when id=id' -> Some occs | _::l -> hyp_occ l in match cls.onhyps with None -> Some [] | Some l -> hyp_occ l let occurrences_of_goal cls = if cls.onconcl then Some cls.concl_occs else None let in_every_hyp cls = (cls.onhyps=None) (* (* Implementation with generalisation then re-intro: introduces noise *) (* in proofs *) let letin_abstract id c occs gl = let env = pf_env gl in let compute_dependency _ (hyp,_,_ as d) ctxt = let d' = try match occurrences_of_hyp hyp occs with | None -> raise Not_found | Some occ -> let newdecl = subst_term_occ_decl occ c d in if occ = [] & d = newdecl then if not (in_every_hyp occs) then raise (RefinerError (DoesNotOccurIn (c,hyp))) else raise Not_found else (subst1_decl (mkVar id) newdecl, true) with Not_found -> (d,List.exists (fun ((id,_,_),dep) -> dep && occur_var_in_decl env id d) ctxt) in d'::ctxt in let ctxt' = fold_named_context compute_dependency env ~init:[] in let compute_marks ((depdecls,marks as accu),lhyp) ((hyp,_,_) as d,b) = if b then ((d::depdecls,(hyp,lhyp)::marks), lhyp) else (accu, Some hyp) in let (depdecls,marks),_ = List.fold_left compute_marks (([],[]),None) ctxt' in let ccl = match occurrences_of_goal occs with | None -> pf_concl gl | Some occ -> subst1 (mkVar id) (subst_term_occ occ c (pf_concl gl)) in (depdecls,marks,ccl) let letin_tac with_eq name c occs gl = let x = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c) name in let id = if name = Anonymous then fresh_id [] x gl else if not (mem_named_context x (pf_hyps gl)) then x else error ("The variable "^(string_of_id x)^" is already declared") in let (depdecls,marks,ccl)= letin_abstract id c occs gl in let t = pf_type_of gl c in let tmpcl = List.fold_right mkNamedProd_or_LetIn depdecls ccl in let args = Array.to_list (instance_from_named_context depdecls) in let newcl = mkNamedLetIn id c t tmpcl in let lastlhyp = if marks=[] then None else snd (List.hd marks) in tclTHENLIST [ apply_type newcl args; thin (List.map (fun (id,_,_) -> id) depdecls); intro_gen (IntroMustBe id) lastlhyp false; if with_eq then tclIDTAC else thin_body [id]; intros_move marks ] gl *) (* Implementation without generalisation: abbrev will be lost in hyps in *) (* in the extracted proof *) let letin_abstract id c occs gl = let env = pf_env gl in let compute_dependency _ (hyp,_,_ as d) depdecls = match occurrences_of_hyp hyp occs with | None -> depdecls | Some occ -> let newdecl = subst_term_occ_decl occ c d in if occ = [] & d = newdecl then if not (in_every_hyp occs) then raise (RefinerError (DoesNotOccurIn (c,hyp))) else depdecls else (subst1_decl (mkVar id) newdecl)::depdecls in let depdecls = fold_named_context compute_dependency env ~init:[] in let ccl = match occurrences_of_goal occs with | None -> pf_concl gl | Some occ -> subst1 (mkVar id) (subst_term_occ occ c (pf_concl gl)) in let lastlhyp = if depdecls = [] then None else Some(pi1(list_last depdecls)) in (depdecls,lastlhyp,ccl) let letin_tac with_eq name c occs gl = let id = let x = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c) name in if name = Anonymous then fresh_id [] x gl else if not (mem_named_context x (pf_hyps gl)) then x else error ("The variable "^(string_of_id x)^" is already declared") in let (depdecls,lastlhyp,ccl)= letin_abstract id c occs gl in let t = refresh_universes (pf_type_of gl c) in let newcl = mkNamedLetIn id c t ccl in tclTHENLIST [ convert_concl_no_check newcl DEFAULTcast; intro_gen (IntroMustBe id) lastlhyp true; if with_eq then tclIDTAC else thin_body [id]; tclMAP convert_hyp_no_check depdecls ] gl let check_hypotheses_occurrences_list env (_,occl) = let rec check acc = function | (hyp,_) :: rest -> if List.mem hyp acc then error ("Hypothesis "^(string_of_id hyp)^" occurs twice"); if not (mem_named_context hyp (named_context env)) then error ("No such hypothesis: " ^ (string_of_id hyp)); check (hyp::acc) rest | [] -> () in check [] occl let nowhere = {onhyps=Some[]; onconcl=false; concl_occs=[]} (* Tactic Assert (b=false) and Pose (b=true): the behaviour of Pose is corrected by the translator. not that of Assert *) let forward b na c = let wh = if !Options.v7 && b then onConcl else nowhere in letin_tac b na c wh (********************************************************************) (* Exact tactics *) (********************************************************************) let exact_check c gl = let concl = (pf_concl gl) in let ct = pf_type_of gl c in if pf_conv_x_leq gl ct concl then refine_no_check c gl else error "Not an exact proof" let exact_no_check = refine_no_check let exact_proof c gl = (* on experimente la synthese d'ise dans exact *) let c = Constrintern.interp_casted_constr (project gl) (pf_env gl) c (pf_concl gl) in refine_no_check c gl let (assumption : tactic) = fun gl -> let concl = pf_concl gl in let hyps = pf_hyps gl in let rec arec only_eq = function | [] -> if only_eq then arec false hyps else error "No such assumption" | (id,c,t)::rest -> if (only_eq & eq_constr t concl) or (not only_eq & pf_conv_x_leq gl t concl) then refine_no_check (mkVar id) gl else arec only_eq rest in arec true hyps (*****************************************************************) (* Modification of a local context *) (*****************************************************************) (* This tactic enables the user to remove hypotheses from the signature. * Some care is taken to prevent him from removing variables that are * subsequently used in other hypotheses or in the conclusion of the * goal. *) let clear ids gl = (* avant seul dyn_clear n'echouait pas en [] *) if ids=[] then tclIDTAC gl else with_check (thin ids) gl let clear_body = thin_body (* Takes a list of booleans, and introduces all the variables * quantified in the goal which are associated with a value * true in the boolean list. *) let rec intros_clearing = function | [] -> tclIDTAC | (false::tl) -> tclTHEN intro (intros_clearing tl) | (true::tl) -> tclTHENLIST [ intro; onLastHyp (fun id -> clear [id]); intros_clearing tl] (* Adding new hypotheses *) let new_hyp mopt (c,lbind) g = let clause = make_clenv_binding g (c,pf_type_of g c) lbind in let (thd,tstack) = whd_stack (clenv_value clause) in let nargs = List.length tstack in let cut_pf = applist(thd, match mopt with | Some m -> if m < nargs then list_firstn m tstack else tstack | None -> tstack) in (tclTHENLAST (tclTHEN (tclEVARS (evars_of clause.env)) (cut (pf_type_of g cut_pf))) ((tclORELSE (apply cut_pf) (exact_no_check cut_pf)))) g (* Keeping only a few hypotheses *) let keep hyps gl = let env = Global.env() in let ccl = pf_concl gl in let cl,_ = fold_named_context_reverse (fun (clear,keep) (hyp,_,_ as decl) -> if List.mem hyp hyps or List.exists (occur_var_in_decl env hyp) keep or occur_var env hyp ccl then (clear,decl::keep) else (hyp::clear,keep)) ~init:([],[]) (pf_env gl) in thin cl gl (************************) (* Introduction tactics *) (************************) let constructor_tac boundopt i lbind gl = let cl = pf_concl gl in let (mind,redcl) = pf_reduce_to_quantified_ind gl cl in let nconstr = Array.length (snd (Global.lookup_inductive mind)).mind_consnames in if i=0 then error "The constructors are numbered starting from 1"; if i > nconstr then error "Not enough constructors"; begin match boundopt with | Some expctdnum -> if expctdnum <> nconstr then error "Not the expected number of constructors" | None -> () end; let cons = mkConstruct (ith_constructor_of_inductive mind i) in let apply_tac = apply_with_bindings (cons,lbind) in (tclTHENLIST [convert_concl_no_check redcl DEFAULTcast; intros; apply_tac]) gl let one_constructor i = constructor_tac None i (* Try to apply the constructor of the inductive definition followed by a tactic t given as an argument. Should be generalize in Constructor (Fun c : I -> tactic) *) let any_constructor tacopt gl = let t = match tacopt with None -> tclIDTAC | Some t -> t in let mind = fst (pf_reduce_to_quantified_ind gl (pf_concl gl)) in let nconstr = Array.length (snd (Global.lookup_inductive mind)).mind_consnames in if nconstr = 0 then error "The type has no constructors"; tclFIRST (List.map (fun i -> tclTHEN (one_constructor i NoBindings) t) (interval 1 nconstr)) gl let left = constructor_tac (Some 2) 1 let simplest_left = left NoBindings let right = constructor_tac (Some 2) 2 let simplest_right = right NoBindings let split = constructor_tac (Some 1) 1 let simplest_split = split NoBindings (********************************************) (* Elimination tactics *) (********************************************) let last_arg c = match kind_of_term c with | App (f,cl) -> array_last cl | _ -> anomaly "last_arg" let elimination_clause_scheme allow_K elimclause indclause gl = let indmv = (match kind_of_term (last_arg elimclause.templval.rebus) with | Meta mv -> mv | _ -> errorlabstrm "elimination_clause" (str "The type of elimination clause is not well-formed")) in let elimclause' = clenv_fchain indmv elimclause indclause in res_pf elimclause' ~allow_K:allow_K gl (* cast added otherwise tactics Case (n1,n2) generates (?f x y) and * refine fails *) let type_clenv_binding wc (c,t) lbind = clenv_type (make_clenv_binding wc (c,t) lbind) (* * Elimination tactic with bindings and using an arbitrary * elimination constant called elimc. This constant should end * with a clause (x:I)(P .. ), where P is a bound variable. * The term c is of type t, which is a product ending with a type * matching I, lbindc are the expected terms for c arguments *) let general_elim_clause elimtac (c,lbindc) (elimc,lbindelimc) gl = let ct = pf_type_of gl c in let t = try snd (pf_reduce_to_quantified_ind gl ct) with UserError _ -> ct in let indclause = make_clenv_binding gl (c,t) lbindc in let elimt = pf_type_of gl elimc in let elimclause = make_clenv_binding gl (elimc,elimt) lbindelimc in elimtac elimclause indclause gl let general_elim c e ?(allow_K=true) = general_elim_clause (elimination_clause_scheme allow_K) c e (* Elimination tactic with bindings but using the default elimination * constant associated with the type. *) let find_eliminator c gl = let (ind,t) = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in Indrec.lookup_eliminator ind (elimination_sort_of_goal gl) let default_elim (c,_ as cx) gl = general_elim cx (find_eliminator c gl,NoBindings) gl let elim_in_context c = function | Some elim -> general_elim c elim ~allow_K:true | None -> default_elim c let elim (c,lbindc as cx) elim = match kind_of_term c with | Var id when lbindc = NoBindings -> tclTHEN (tclTRY (intros_until_id id)) (elim_in_context cx elim) | _ -> elim_in_context cx elim (* The simplest elimination tactic, with no substitutions at all. *) let simplest_elim c = default_elim (c,NoBindings) (* Elimination in hypothesis *) let elimination_in_clause_scheme id elimclause indclause gl = let (hypmv,indmv) = match clenv_independent elimclause with [k1;k2] -> (k1,k2) | _ -> errorlabstrm "elimination_clause" (str "The type of elimination clause is not well-formed") in let elimclause' = clenv_fchain indmv elimclause indclause in let hyp = mkVar id in let hyp_typ = pf_type_of gl hyp in let hypclause = mk_clenv_from_n gl (Some 0) (hyp, hyp_typ) in let elimclause'' = clenv_fchain hypmv elimclause' hypclause in let new_hyp_prf = clenv_value elimclause'' in let new_hyp_typ = clenv_type elimclause'' in if eq_constr hyp_typ new_hyp_typ then errorlabstrm "general_rewrite_in" (str "Nothing to rewrite in " ++ pr_id id); tclTHEN (tclEVARS (evars_of elimclause''.env)) (cut_replacing id new_hyp_typ (fun x gls -> refine_no_check new_hyp_prf gls)) gl let general_elim_in id = general_elim_clause (elimination_in_clause_scheme id) (* Case analysis tactics *) let general_case_analysis_in_context (c,lbindc) gl = let (mind,_) = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in let sort = elimination_sort_of_goal gl in let case = if occur_term c (pf_concl gl) then make_case_dep else make_case_gen in let elim = pf_apply case gl mind sort in general_elim (c,lbindc) (elim,NoBindings) gl let general_case_analysis (c,lbindc as cx) = match kind_of_term c with | Var id when lbindc = NoBindings -> tclTHEN (tclTRY (intros_until_id id)) (general_case_analysis_in_context cx) | _ -> general_case_analysis_in_context cx let simplest_case c = general_case_analysis (c,NoBindings) (*****************************) (* Decomposing introductions *) (*****************************) let clear_last = tclLAST_HYP (fun c -> (clear [destVar c])) let case_last = tclLAST_HYP simplest_case let rec explicit_intro_names = function | IntroWildcard :: l -> explicit_intro_names l | IntroIdentifier id :: l -> id :: explicit_intro_names l | IntroOrAndPattern ll :: l' -> List.flatten (List.map (fun l -> explicit_intro_names (l@l')) ll) | [] -> [] (* We delay thinning until the completion of the whole intros tactic to ensure that dependent hypotheses are cleared in the right dependency order (see bug #1000); we use fresh names, not used in the tactic, for the hyps to clear *) let rec intros_patterns thin destopt = function | IntroWildcard :: l -> tclTHEN (intro_gen (IntroAvoid (explicit_intro_names l)) None true) (onLastHyp (fun id -> intros_patterns (id::thin) destopt l)) | IntroIdentifier id :: l -> tclTHEN (intro_gen (IntroMustBe id) destopt true) (intros_patterns thin destopt l) | IntroOrAndPattern ll :: l' -> tclTHEN introf (tclTHENS (tclTHEN case_last clear_last) (List.map (fun l -> intros_patterns thin destopt (l@l')) ll)) | [] -> clear thin let intros_pattern = intros_patterns [] let intro_pattern destopt pat = intros_patterns [] destopt [pat] let intro_patterns = function | [] -> tclREPEAT intro | l -> intros_pattern None l (* * A "natural" induction tactic * - [H0:T0, ..., Hi:Ti, hyp0:P->I(args), Hi+1:Ti+1, ..., Hn:Tn |-G] is the goal - [hyp0] is the induction hypothesis - we extract from [args] the variables which are not rigid parameters of the inductive type, this is [indvars] (other terms are forgotten); [indhyps] are the ones which actually are declared in context (done in [find_atomic_param_of_ind]) - we look for all hyps depending of [hyp0] or one of [indvars]: this is [dephyps] of types [deptyps] respectively - [statuslist] tells for each hyps in [dephyps] after which other hyp fixed in the context they must be moved (when induction is done) - [hyp0succ] is the name of the hyp fixed in the context after which to move the subterms of [hyp0succ] in the i-th branch where it is supposed to be the i-th constructor of the inductive type. Strategy: (cf in [induction_from_context]) - requantify and clear all [dephyps] - apply induction on [hyp0] - clear [indhyps] and [hyp0] - in the i-th subgoal, intro the arguments of the i-th constructor of the inductive type after [hyp0succ] (done in [induct_discharge]) let the induction hypotheses on top of the hyps because they may depend on variables between [hyp0] and the top. A counterpart is that the dep hyps programmed to be intro-ed on top must now be intro-ed after the induction hypotheses - move each of [dephyps] at the right place following the [statuslist] *) let rec str_intro_pattern = function | IntroOrAndPattern pll -> "["^(String.concat "|" (List.map (fun pl -> String.concat " " (List.map str_intro_pattern pl)) pll)) ^"]" | IntroWildcard -> "_" | IntroIdentifier id -> string_of_id id let check_unused_names names = if names <> [] & Options.is_verbose () then let s = if List.tl names = [] then " " else "s " in let names = String.concat " " (List.map str_intro_pattern names) in warning ("Unused introduction pattern"^s^": "^names) let rec first_name_buggy = function | IntroOrAndPattern [] -> None | IntroOrAndPattern ([]::l) -> first_name_buggy (IntroOrAndPattern l) | IntroOrAndPattern ((p::_)::_) -> first_name_buggy p | IntroWildcard -> None | IntroIdentifier id -> Some id type elim_arg_kind = RecArg | IndArg | OtherArg let induct_discharge statuslists destopt avoid' ((avoid7,avoid8),ra) (names,force,rnames) gl = let avoid7 = avoid7 @ avoid' in let avoid8 = avoid8 @ avoid' in let (lstatus,rstatus) = statuslists in let tophyp = ref None in let rec peel_tac ra names gl = match ra with | (RecArg,(recvarname7,recvarname8)) :: (IndArg,(hyprecname7,hyprecname8)) :: ra' -> let recpat,hyprec,names = match names with | [] -> let idrec7 = (fresh_id avoid7 recvarname7 gl) in let idrec8 = (fresh_id avoid8 recvarname8 gl) in let idhyp7 = (fresh_id avoid7 hyprecname7 gl) in let idhyp8 = (fresh_id avoid8 hyprecname8 gl) in if Options.do_translate() & (idrec7 <> idrec8 or idhyp7 <> idhyp8) then force := true; let idrec = if !Options.v7 then idrec7 else idrec8 in let idhyp = if !Options.v7 then idhyp7 else idhyp8 in (IntroIdentifier idrec, IntroIdentifier idhyp, []) | [IntroIdentifier id as pat] -> let id7 = next_ident_away (add_prefix "IH" id) avoid7 in let id8 = next_ident_away (add_prefix "IH" id) avoid8 in if Options.do_translate() & id7 <> id8 then force := true; let id = if !Options.v7 then id7 else id8 in (pat, IntroIdentifier id, []) | [pat] -> let idhyp7 = (fresh_id avoid7 hyprecname7 gl) in let idhyp8 = (fresh_id avoid8 hyprecname8 gl) in if Options.do_translate() & idhyp7 <> idhyp8 then force := true; let idhyp = if !Options.v7 then idhyp7 else idhyp8 in (pat, IntroIdentifier idhyp, []) | pat1::pat2::names -> (pat1,pat2,names) in (* This is buggy for intro-or-patterns with different first hypnames *) if !tophyp=None then tophyp := first_name_buggy hyprec; rnames := !rnames @ [recpat; hyprec]; tclTHENLIST [ intros_pattern destopt [recpat]; intros_pattern None [hyprec]; peel_tac ra' names ] gl | (IndArg,(hyprecname7,hyprecname8)) :: ra' -> (* Rem: does not happen in Coq schemes, only in user-defined schemes *) let pat,names = match names with | [] -> IntroIdentifier (fresh_id avoid8 hyprecname8 gl), [] | pat::names -> pat,names in rnames := !rnames @ [pat]; tclTHEN (intros_pattern destopt [pat]) (peel_tac ra' names) gl | (RecArg,(recvarname7,recvarname8)) :: ra' -> let introtac,names = match names with | [] -> let id8 = fresh_id avoid8 recvarname8 gl in let i = if !Options.v7 then IntroAvoid avoid7 else IntroMustBe id8 in (* For translator *) let id7 = fresh_id avoid7 (default_id gl (match kind_of_term (pf_concl gl) with | Prod (name,t,_) -> (name,None,t) | LetIn (name,b,t,_) -> (name,Some b,t) | _ -> raise (RefinerError IntroNeedsProduct))) gl in if Options.do_translate() & id7 <> id8 then force := true; let id = if !Options.v7 then id7 else id8 in rnames := !rnames @ [IntroIdentifier id]; intro_gen i destopt false, [] | pat::names -> rnames := !rnames @ [pat]; intros_pattern destopt [pat],names in tclTHEN introtac (peel_tac ra' names) gl | (OtherArg,_) :: ra' -> let introtac,names = match names with | [] -> (* For translator *) let id7 = fresh_id avoid7 (default_id gl (match kind_of_term (pf_concl gl) with | Prod (name,t,_) -> (name,None,t) | LetIn (name,b,t,_) -> (name,Some b,t) | _ -> raise (RefinerError IntroNeedsProduct))) gl in let id8 = fresh_id avoid8 (default_id gl (match kind_of_term (pf_concl gl) with | Prod (name,t,_) -> (name,None,t) | LetIn (name,b,t,_) -> (name,Some b,t) | _ -> raise (RefinerError IntroNeedsProduct))) gl in if Options.do_translate() & id7 <> id8 then force := true; let id = if !Options.v7 then id7 else id8 in let avoid = if !Options.v7 then avoid7 else avoid8 in rnames := !rnames @ [IntroIdentifier id]; intro_gen (IntroAvoid avoid) destopt false, [] | pat::names -> rnames := !rnames @ [pat]; intros_pattern destopt [pat],names in tclTHEN introtac (peel_tac ra' names) gl | [] -> check_unused_names names; tclIDTAC gl in let intros_move lstatus = let newlstatus = (* if some IH has taken place at the top of hyps *) List.map (function (hyp,None) -> (hyp,!tophyp) | x -> x) lstatus in intros_move newlstatus in tclTHENLIST [ peel_tac ra names; intros_rmove rstatus; intros_move lstatus ] gl (* - le recalcul de indtyp à chaque itération de atomize_one est pour ne pas s'embêter à regarder si un letin_tac ne fait pas des substitutions aussi sur l'argument voisin *) (* Marche pas... faut prendre en compte l'occurrence précise... *) let atomize_param_of_ind (indref,nparams) hyp0 gl = let tmptyp0 = pf_get_hyp_typ gl hyp0 in let typ0 = pf_apply reduce_to_quantified_ref gl indref tmptyp0 in let prods, indtyp = decompose_prod typ0 in let argl = snd (decompose_app indtyp) in let params = list_firstn nparams argl in (* le gl est important pour ne pas préévaluer *) let rec atomize_one i avoid gl = if i<>nparams then let tmptyp0 = pf_get_hyp_typ gl hyp0 in (* If argl <> [], we expect typ0 not to be quantified, in order to avoid bound parameters... then we call pf_reduce_to_atomic_ind *) let indtyp = pf_apply reduce_to_atomic_ref gl indref tmptyp0 in let argl = snd (decompose_app indtyp) in let c = List.nth argl (i-1) in match kind_of_term c with | Var id when not (List.exists (occur_var (pf_env gl) id) avoid) -> atomize_one (i-1) ((mkVar id)::avoid) gl | Var id -> let x = fresh_id [] id gl in tclTHEN (letin_tac true (Name x) (mkVar id) allClauses) (atomize_one (i-1) ((mkVar x)::avoid)) gl | _ -> let id = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c) Anonymous in let x = fresh_id [] id gl in tclTHEN (letin_tac true (Name x) c allClauses) (atomize_one (i-1) ((mkVar x)::avoid)) gl else tclIDTAC gl in atomize_one (List.length argl) params gl let find_atomic_param_of_ind nparams indtyp = let argl = snd (decompose_app indtyp) in let argv = Array.of_list argl in let params = list_firstn nparams argl in let indvars = ref Idset.empty in for i = nparams to (Array.length argv)-1 do match kind_of_term argv.(i) with | Var id when not (List.exists (occur_var (Global.env()) id) params) -> indvars := Idset.add id !indvars | _ -> () done; Idset.elements !indvars; (* [cook_sign] builds the lists [indhyps] of hyps that must be erased, the lists of hyps to be generalize [(hdeps,tdeps)] on the goal together with the places [(lstatus,rstatus)] where to re-intro them after induction. To know where to re-intro the dep hyp, we remember the name of the hypothesis [lhyp] after which (if the dep hyp is more recent than [hyp0]) or [rhyp] before which (if older than [hyp0]) its equivalent must be moved when the induction has been applied. Since computation of dependencies and [rhyp] is from more ancient (on the right) to more recent hyp (on the left) but the computation of [lhyp] progresses from the other way, [cook_hyp] is in two passes (an alternative would have been to write an higher-order algorithm). We strongly use references to reduce the accumulation of arguments. To summarize, the situation looks like this Goal(n,x) -| H6:(Q n); x:A; H5:True; H4:(le O n); H3:(P n); H2:True; n:nat Left Right Induction hypothesis is H4 ([hyp0]) Variable parameters of (le O n) is the singleton list with "n" ([indvars]) Part of [indvars] really in context is the same ([indhyps]) The dependent hyps are H3 and H6 ([dephyps]) For H3 the memorized places are H5 ([lhyp]) and H2 ([rhyp]) because these names are among the hyp which are fixed through the induction For H6 the neighbours are None ([lhyp]) and H5 ([rhyp]) For H3, because on the right of H4, we remember rhyp (here H2) For H6, because on the left of H4, we remember lhyp (here None) For H4, we remember lhyp (here H5) The right neighbour is then translated into the left neighbour because move_hyp tactic needs the name of the hyp _after_ which we move the hyp to move. But, say in the 2nd subgoal of the hypotheses, the goal will be (m:nat)((P m)->(Q m)->(Goal m)) -> (P Sm)-> (Q Sm)-> (Goal Sm) ^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^ both go where H4 was goes where goes where H3 was H6 was We have to intro and move m and the recursive hyp first, but then where to move H3 ??? Only the hyp on its right is relevant, but we have to translate it into the name of the hyp on the left Note: this case where some hyp(s) in [dephyps] has(have) the same left neighbour as [hyp0] is the only problematic case with right neighbours. For the other cases (e.g. an hyp H1:(R n) between n and H2 would have posed no problem. But for uniformity, we decided to use the right hyp for all hyps on the right of H4. Others solutions are welcome *) exception Shunt of identifier option let cook_sign hyp0 indvars env = (* First phase from L to R: get [indhyps], [decldep] and [statuslist] for the hypotheses before (= more ancient than) hyp0 (see above) *) let allindhyps = hyp0::indvars in let indhyps = ref [] in let decldeps = ref [] in let ldeps = ref [] in let rstatus = ref [] in let lstatus = ref [] in let before = ref true in let seek_deps env (hyp,_,_ as decl) rhyp = if hyp = hyp0 then begin before:=false; None (* fake value *) end else if List.mem hyp indvars then begin (* warning: hyp can still occur after induction *) (* e.g. if the goal (t hyp hyp0) with other occs of hyp in t *) indhyps := hyp::!indhyps; rhyp end else if (List.exists (fun id -> occur_var_in_decl env id decl) allindhyps or List.exists (fun (id,_,_) -> occur_var_in_decl env id decl) !decldeps) then begin decldeps := decl::!decldeps; if !before then rstatus := (hyp,rhyp)::!rstatus else ldeps := hyp::!ldeps; (* status computed in 2nd phase *) Some hyp end else Some hyp in let _ = fold_named_context seek_deps env ~init:None in (* 2nd phase from R to L: get left hyp of [hyp0] and [lhyps] *) let compute_lstatus lhyp (hyp,_,_) = if hyp = hyp0 then raise (Shunt lhyp); if List.mem hyp !ldeps then begin lstatus := (hyp,lhyp)::!lstatus; lhyp end else if List.mem hyp !indhyps then lhyp else (Some hyp) in try let _ = fold_named_context_reverse compute_lstatus ~init:None env in anomaly "hyp0 not found" with Shunt lhyp0 -> let statuslists = (!lstatus,List.rev !rstatus) in (statuslists, lhyp0, !indhyps, !decldeps) let induction_tac varname typ ((elimc,lbindelimc),elimt) gl = let c = mkVar varname in let indclause = make_clenv_binding gl (c,typ) NoBindings in let elimclause = make_clenv_binding gl (mkCast (elimc,DEFAULTcast, elimt),elimt) lbindelimc in elimination_clause_scheme true elimclause indclause gl let make_up_names7 n ind (old_style,cname) = if old_style (* = V6.3 version of Induction on hypotheses *) then let recvarname = if n=1 then cname else (* To force renumbering if there is only one *) make_ident (string_of_id cname ) (Some 1) in recvarname, add_prefix "Hrec" recvarname, [] else let is_hyp = atompart_of_id cname = "H" in let hyprecname = add_prefix "IH" (if is_hyp then Nametab.id_of_global ind else cname) in let avoid = if n=1 (* Only one recursive argument *) or (* Rem: no recursive argument (especially if Destruct) *) n=0 (* & atompart_of_id cname <> "H" (* for 7.1 compatibility *)*) then [] else (* Forbid to use cname, cname0, hyprecname and hyprecname0 *) (* in order to get names such as f1, f2, ... *) let avoid = (make_ident (string_of_id cname) (Some 0)) ::(*here for 7.1 cmpat*) (make_ident (string_of_id hyprecname) None) :: (make_ident (string_of_id hyprecname) (Some 0)) :: [] in if atompart_of_id cname <> "H" then (make_ident (string_of_id cname) None) :: avoid else avoid in cname, hyprecname, avoid let make_base n id = if n=0 or n=1 then id else (* This extends the name to accept new digits if it already ends with *) (* digits *) id_of_string (atompart_of_id (make_ident (string_of_id id) (Some 0))) let make_up_names8 n ind (_,cname) = let is_hyp = atompart_of_id cname = "H" in let base = string_of_id (make_base n cname) in let hyprecname = add_prefix "IH" (make_base n (if is_hyp then Nametab.id_of_global ind else cname)) in let avoid = if n=1 (* Only one recursive argument *) or n=0 then [] else (* Forbid to use cname, cname0, hyprecname and hyprecname0 *) (* in order to get names such as f1, f2, ... *) let avoid = (make_ident (string_of_id hyprecname) None) :: (make_ident (string_of_id hyprecname) (Some 0)) :: [] in if atompart_of_id cname <> "H" then (make_ident base (Some 0)) :: (make_ident base None) :: avoid else avoid in id_of_string base, hyprecname, avoid let is_indhyp p n t = let l, c = decompose_prod t in let c,_ = decompose_app c in let p = p + List.length l in match kind_of_term c with | Rel k when p < k & k <= p + n -> true | _ -> false let chop_context n l = let rec chop_aux acc = function | n, (_,Some _,_ as h :: t) -> chop_aux (h::acc) (n, t) | 0, l2 -> (List.rev acc, l2) | n, (h::t) -> chop_aux (h::acc) (n-1, t) | _, [] -> anomaly "chop_context" in chop_aux [] (n,l) let error_ind_scheme s = let s = if s <> "" then s^" " else s in error ("Cannot recognise "^s^"an induction schema") (* Check that the elimination scheme has a form similar to the elimination schemes built by Coq *) let compute_elim_signature elimt names_info = let hyps,ccl = decompose_prod_assum elimt in let n = List.length hyps in if n = 0 then error_ind_scheme ""; let f,l = decompose_app ccl in let _,indbody,ind = List.hd hyps in if indbody <> None then error "Cannot recognise an induction scheme"; let nargs = List.length l in let dep = (nargs >= 1 && list_last l = mkRel 1) in let nrealargs = if dep then nargs-1 else nargs in let args = if dep then list_firstn nrealargs l else l in let realargs,hyps1 = chop_context nrealargs (List.tl hyps) in if args <> extended_rel_list 1 realargs then error_ind_scheme "the conclusion of"; let indhd,indargs = decompose_app ind in let indt = try global_of_constr indhd with _ -> error "Cannot find the inductive type of the inductive schema" in let nparams = List.length indargs - nrealargs in let revparams, revhyps2 = chop_context nparams (List.rev hyps1) in let rec check_elim npred = function | (na,None,t)::l when isSort (snd (decompose_prod_assum t)) -> check_elim (npred+1) l | l -> let is_pred n c = let hd = fst (decompose_app c) in match kind_of_term hd with | Rel q when n < q & q <= n+npred -> IndArg | _ when hd = indhd -> RecArg | _ -> OtherArg in let rec check_branch p c = match kind_of_term c with | Prod (_,t,c) -> is_pred p t :: check_branch (p+1) c | LetIn (_,_,_,c) -> OtherArg :: check_branch (p+1) c (* | App (f,_) when is_pred p f = IndArg -> []*) | _ when is_pred p c = IndArg -> [] | _ -> raise Exit in let rec find_branches p = function | (_,None,t)::brs -> (match try Some (check_branch p t) with Exit -> None with | Some l -> let n7 = List.fold_left (fun n b -> if b=IndArg then n+1 else n) 0 l in let n8 = List.fold_left (fun n b -> if b=RecArg then n+1 else n) 0 l in let recvarname7, hyprecname7, avoid7 = make_up_names7 n7 indt names_info in let recvarname8, hyprecname8, avoid8 = make_up_names8 n8 indt names_info in let namesign = List.map (fun b -> (b,if b=IndArg then (hyprecname7,hyprecname8) else (recvarname7,recvarname8))) l in ((avoid7,avoid8),namesign) :: find_branches (p+1) brs | None -> error_ind_scheme "the branches of") | (_,Some _,_)::_ -> error_ind_scheme "the branches of" | [] -> (* Check again conclusion *) let ccl_arg_ok = is_pred (p + List.length realargs + 1) f = IndArg in let ind_is_ok = list_lastn nrealargs indargs = extended_rel_list 0 realargs in if not (ccl_arg_ok & ind_is_ok) then error "Cannot recognize the conclusion of an induction schema"; [] in find_branches 0 l in nparams, indt, (Array.of_list (check_elim 0 revhyps2)) let find_elim_signature isrec style elim hyp0 gl = let tmptyp0 = pf_get_hyp_typ gl hyp0 in let (elimc,elimt) = match elim with | None -> let mind,_ = pf_reduce_to_quantified_ind gl tmptyp0 in let s = elimination_sort_of_goal gl in let elimc = if isrec then Indrec.lookup_eliminator mind s else pf_apply Indrec.make_case_gen gl mind s in let elimt = pf_type_of gl elimc in ((elimc, NoBindings), elimt) | Some (elimc,lbind as e) -> (e, pf_type_of gl elimc) in let name_info = (style,hyp0) in let nparams,indref,indsign = compute_elim_signature elimt name_info in (elimc,elimt,nparams,indref,indsign) let induction_from_context isrec elim_info hyp0 (names,b_rnames) gl = (*test suivant sans doute inutile car refait par le letin_tac*) if List.mem hyp0 (ids_of_named_context (Global.named_context())) then errorlabstrm "induction" (str "Cannot generalize a global variable"); let elimc,elimt,nparams,indref,indsign = elim_info in let tmptyp0 = pf_get_hyp_typ gl hyp0 in let typ0 = pf_apply reduce_to_quantified_ref gl indref tmptyp0 in let env = pf_env gl in let indvars = find_atomic_param_of_ind nparams (snd (decompose_prod typ0)) in let (statlists,lhyp0,indhyps,deps) = cook_sign hyp0 indvars env in let tmpcl = it_mkNamedProd_or_LetIn (pf_concl gl) deps in let names = compute_induction_names (Array.length indsign) names in (* For translator *) let names' = Array.map ref (Array.make (Array.length indsign) []) in let b = ref false in b_rnames := (b,Array.to_list names')::!b_rnames; let names = array_map2 (fun n n' -> (n,b,n')) names names' in (* End translator *) let dephyps = List.map (fun (id,_,_) -> id) deps in let args = List.fold_left (fun a (id,b,_) -> if b = None then (mkVar id)::a else a) [] deps in (* Magistral effet de bord: si hyp0 a des arguments, ceux d'entre eux qui ouvrent de nouveaux buts arrivent en premier dans la liste des sous-buts du fait qu'ils sont le plus à gauche dans le combinateur engendré par make_case_gen (un "Cases (hyp0 ?) of ...") et il faut alors appliquer tclTHENLASTn; en revanche, comme lookup_eliminator renvoie un combinateur de la forme "ind_rec ... (hyp0 ?)", les buts correspondant à des arguments de hyp0 sont maintenant à la fin et c'est tclTHENFIRSTn qui marche !!! *) tclTHENLIST [ if deps = [] then tclIDTAC else apply_type tmpcl args; thin dephyps; (if isrec then tclTHENFIRSTn else tclTHENLASTn) (tclTHENLIST [ induction_tac hyp0 typ0 (elimc,elimt); thin [hyp0]; tclTRY (thin indhyps) ]) (array_map2 (induct_discharge statlists lhyp0 (List.rev dephyps)) indsign names) ] gl let induction_with_atomization_of_ind_arg isrec elim names hyp0 gl = let (elimc,elimt,nparams,indref,indsign as elim_info) = find_elim_signature isrec false elim hyp0 gl in tclTHEN (atomize_param_of_ind (indref,nparams) hyp0) (induction_from_context isrec elim_info hyp0 names) gl (* This is Induction since V7 ("natural" induction both in quantified premisses and introduced ones) *) let new_induct_gen isrec elim names c gl = match kind_of_term c with | Var id when not (mem_named_context id (Global.named_context())) -> induction_with_atomization_of_ind_arg isrec elim names id gl | _ -> let x = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c) Anonymous in let id = fresh_id [] x gl in tclTHEN (letin_tac true (Name id) c allClauses) (induction_with_atomization_of_ind_arg isrec elim names id) gl let new_induct_destruct isrec c elim names = match c with | ElimOnConstr c -> new_induct_gen isrec elim names c | ElimOnAnonHyp n -> tclTHEN (intros_until_n n) (tclLAST_HYP (new_induct_gen isrec elim names)) (* Identifier apart because id can be quantified in goal and not typable *) | ElimOnIdent (_,id) -> tclTHEN (tclTRY (intros_until_id id)) (new_induct_gen isrec elim names (mkVar id)) let new_induct = new_induct_destruct true let new_destruct = new_induct_destruct false (* The registered tactic, which calls the default elimination * if no elimination constant is provided. *) (* Induction tactics *) (* This was Induction before 6.3 (induction only in quantified premisses) *) let raw_induct s = tclTHEN (intros_until_id s) (tclLAST_HYP simplest_elim) let raw_induct_nodep n = tclTHEN (intros_until_n n) (tclLAST_HYP simplest_elim) (* This was Induction in 6.3 (hybrid form) *) let induction_from_context_old_style hyp b_ids gl = let elim_info = find_elim_signature true true None hyp gl in let x = induction_from_context true elim_info hyp (None,b_ids) gl in (* For translator *) fst (List.hd !b_ids) := true; x let simple_induct_id hyp b_ids = if !Options.v7 then tclORELSE (raw_induct hyp) (induction_from_context_old_style hyp b_ids) else raw_induct hyp let simple_induct_nodep = raw_induct_nodep let simple_induct = function | NamedHyp id,b_ids -> simple_induct_id id b_ids | AnonHyp n,_ -> simple_induct_nodep n (* Destruction tactics *) let simple_destruct_id s = (tclTHEN (intros_until_id s) (tclLAST_HYP simplest_case)) let simple_destruct_nodep n = (tclTHEN (intros_until_n n) (tclLAST_HYP simplest_case)) let simple_destruct = function | NamedHyp id -> simple_destruct_id id | AnonHyp n -> simple_destruct_nodep n (* * Eliminations giving the type instead of the proof. * These tactics use the default elimination constant and * no substitutions at all. * May be they should be integrated into Elim ... *) let elim_scheme_type elim t gl = let clause = mk_clenv_type_of gl elim in match kind_of_term (last_arg clause.templval.rebus) with | Meta mv -> let clause' = (* t is inductive, then CUMUL or CONV is irrelevant *) clenv_unify true Reduction.CUMUL t (clenv_meta_type clause mv) clause in res_pf clause' ~allow_K:true gl | _ -> anomaly "elim_scheme_type" let elim_type t gl = let (ind,t) = pf_reduce_to_atomic_ind gl t in let elimc = Indrec.lookup_eliminator ind (elimination_sort_of_goal gl) in elim_scheme_type elimc t gl let case_type t gl = let (ind,t) = pf_reduce_to_atomic_ind gl t in let env = pf_env gl in let elimc = Indrec.make_case_gen env (project gl) ind (elimination_sort_of_goal gl) in elim_scheme_type elimc t gl (* Some eliminations frequently used *) (* These elimination tactics are particularly adapted for sequent calculus. They take a clause as argument, and yield the elimination rule if the clause is of the form (Some id) and a suitable introduction rule otherwise. They do not depend on the name of the eliminated constant, so they can be also used on ad-hoc disjunctions and conjunctions introduced by the user. -- Eduardo Gimenez (11/8/97) HH (29/5/99) replaces failures by specific error messages *) let andE id gl = let t = pf_get_hyp_typ gl id in if is_conjunction (pf_hnf_constr gl t) then (tclTHEN (simplest_elim (mkVar id)) (tclDO 2 intro)) gl else errorlabstrm "andE" (str("Tactic andE expects "^(string_of_id id)^" is a conjunction.")) let dAnd cls = onClauses (function | None -> simplest_split | Some (id,_,_) -> andE id) cls let orE id gl = let t = pf_get_hyp_typ gl id in if is_disjunction (pf_hnf_constr gl t) then (tclTHEN (simplest_elim (mkVar id)) intro) gl else errorlabstrm "orE" (str("Tactic orE expects "^(string_of_id id)^" is a disjunction.")) let dorE b cls = onClauses (function | (Some (id,_,_)) -> orE id | None -> (if b then right else left) NoBindings) cls let impE id gl = let t = pf_get_hyp_typ gl id in if is_imp_term (pf_hnf_constr gl t) then let (dom, _, rng) = destProd (pf_hnf_constr gl t) in tclTHENLAST (cut_intro rng) (apply_term (mkVar id) [mkMeta (new_meta())]) gl else errorlabstrm "impE" (str("Tactic impE expects "^(string_of_id id)^ " is a an implication.")) let dImp cls = onClauses (function | None -> intro | Some (id,_,_) -> impE id) cls (************************************************) (* Tactics related with logic connectives *) (************************************************) (* Reflexivity tactics *) let setoid_reflexivity = ref (fun _ -> assert false) let register_setoid_reflexivity f = setoid_reflexivity := f let reflexivity gl = match match_with_equation (pf_concl gl) with | None -> !setoid_reflexivity gl | Some (hdcncl,args) -> one_constructor 1 NoBindings gl let intros_reflexivity = (tclTHEN intros reflexivity) (* Symmetry tactics *) (* This tactic first tries to apply a constant named sym_eq, where eq is the name of the equality predicate. If this constant is not defined and the conclusion is a=b, it solves the goal doing (Cut b=a;Intro H;Case H;Constructor 1) *) let setoid_symmetry = ref (fun _ -> assert false) let register_setoid_symmetry f = setoid_symmetry := f let symmetry gl = match match_with_equation (pf_concl gl) with | None -> !setoid_symmetry gl | Some (hdcncl,args) -> let hdcncls = string_of_inductive hdcncl in begin try (apply (pf_parse_const gl ("sym_"^hdcncls)) gl) with _ -> let symc = match args with | [t1; c1; t2; c2] -> mkApp (hdcncl, [| t2; c2; t1; c1 |]) | [typ;c1;c2] -> mkApp (hdcncl, [| typ; c2; c1 |]) | [c1;c2] -> mkApp (hdcncl, [| c2; c1 |]) | _ -> assert false in tclTHENLAST (cut symc) (tclTHENLIST [ intro; tclLAST_HYP simplest_case; one_constructor 1 NoBindings ]) gl end let setoid_symmetry_in = ref (fun _ _ -> assert false) let register_setoid_symmetry_in f = setoid_symmetry_in := f let symmetry_in id gl = let ctype = pf_type_of gl (mkVar id) in let sign,t = decompose_prod_assum ctype in match match_with_equation t with | None -> !setoid_symmetry_in id gl | Some (hdcncl,args) -> let symccl = match args with | [t1; c1; t2; c2] -> mkApp (hdcncl, [| t2; c2; t1; c1 |]) | [typ;c1;c2] -> mkApp (hdcncl, [| typ; c2; c1 |]) | [c1;c2] -> mkApp (hdcncl, [| c2; c1 |]) | _ -> assert false in tclTHENS (cut (it_mkProd_or_LetIn symccl sign)) [ intro_replacing id; tclTHENLIST [ intros; symmetry; apply (mkVar id); assumption ] ] gl let intros_symmetry = onClauses (function | None -> tclTHEN intros symmetry | Some (id,_,_) -> symmetry_in id) (* Transitivity tactics *) (* This tactic first tries to apply a constant named trans_eq, where eq is the name of the equality predicate. If this constant is not defined and the conclusion is a=b, it solves the goal doing Cut x1=x2; [Cut x2=x3; [Intros e1 e2; Case e2;Assumption | Idtac] | Idtac] --Eduardo (19/8/97) *) let setoid_transitivity = ref (fun _ _ -> assert false) let register_setoid_transitivity f = setoid_transitivity := f let transitivity t gl = match match_with_equation (pf_concl gl) with | None -> !setoid_transitivity t gl | Some (hdcncl,args) -> let hdcncls = string_of_inductive hdcncl in begin try apply_list [(pf_parse_const gl ("trans_"^hdcncls));t] gl with _ -> let eq1, eq2 = match args with | [typ1;c1;typ2;c2] -> let typt = pf_type_of gl t in ( mkApp(hdcncl, [| typ1; c1; typt ;t |]), mkApp(hdcncl, [| typt; t; typ2; c2 |]) ) | [typ;c1;c2] -> ( mkApp (hdcncl, [| typ; c1; t |]), mkApp (hdcncl, [| typ; t; c2 |]) ) | [c1;c2] -> ( mkApp (hdcncl, [| c1; t|]), mkApp (hdcncl, [| t; c2 |]) ) | _ -> assert false in tclTHENFIRST (cut eq2) (tclTHENFIRST (cut eq1) (tclTHENLIST [ tclDO 2 intro; tclLAST_HYP simplest_case; assumption ])) gl end let intros_transitivity n = tclTHEN intros (transitivity n) (* tactical to save as name a subproof such that the generalisation of the current goal, abstracted with respect to the local signature, is solved by tac *) let interpretable_as_section_decl d1 d2 = match d1,d2 with | (_,Some _,_), (_,None,_) -> false | (_,Some b1,t1), (_,Some b2,t2) -> eq_constr b1 b2 & eq_constr t1 t2 | (_,None,t1), (_,_,t2) -> eq_constr t1 t2 let abstract_subproof name tac gls = let current_sign = Global.named_context() and global_sign = pf_hyps gls in let sign,secsign = List.fold_right (fun (id,_,_ as d) (s1,s2) -> if mem_named_context id current_sign & interpretable_as_section_decl (Sign.lookup_named id current_sign) d then (s1,push_named_context_val d s2) else (add_named_decl d s1,s2)) global_sign (empty_named_context,empty_named_context_val) in let na = next_global_ident_away false name (pf_ids_of_hyps gls) in let concl = it_mkNamedProd_or_LetIn (pf_concl gls) sign in if occur_existential concl then if !Options.v7 then error "Abstract cannot handle existentials" else error "\"abstract\" cannot handle existentials"; let lemme = start_proof na (IsGlobal (Proof Lemma)) secsign concl (fun _ _ -> ()); let _,(const,kind,_) = try by (tclCOMPLETE (tclTHEN (tclDO (List.length sign) intro) tac)); let r = cook_proof () in delete_current_proof (); r with e when catchable_exception e -> (delete_current_proof(); raise e) in (* Faudrait un peu fonctionnaliser cela *) let cd = Entries.DefinitionEntry const in let con = Declare.declare_internal_constant na (cd,IsProof Lemma) in constr_of_global (ConstRef con) in exact_no_check (applist (lemme, List.rev (Array.to_list (instance_from_named_context sign)))) gls let tclABSTRACT name_op tac gls = let s = match name_op with | Some s -> s | None -> add_suffix (get_current_proof_name ()) "_subproof" in abstract_subproof s tac gls let admit_as_an_axiom gls = let current_sign = Global.named_context() and global_sign = pf_hyps gls in let sign,secsign = List.fold_right (fun (id,_,_ as d) (s1,s2) -> if mem_named_context id current_sign & interpretable_as_section_decl (Sign.lookup_named id current_sign) d then (s1,add_named_decl d s2) else (add_named_decl d s1,s2)) global_sign (empty_named_context,empty_named_context) in let name = add_suffix (get_current_proof_name ()) "_admitted" in let na = next_global_ident_away false name (pf_ids_of_hyps gls) in let concl = it_mkNamedProd_or_LetIn (pf_concl gls) sign in if occur_existential concl then if !Options.v7 then error "admit cannot handle existentials" else error "\"admit\" cannot handle existentials"; let axiom = let cd = Entries.ParameterEntry concl in let con = Declare.declare_internal_constant na (cd,IsAssumption Logical) in constr_of_global (ConstRef con) in exact_no_check (applist (axiom, List.rev (Array.to_list (instance_from_named_context sign)))) gls