(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* (!strictness)),(fun b -> strictness:=b) let _ = declare_bool_option { optsync = true; optname = "strict mode"; optkey = (SecondaryTable ("Strict","Proofs")); optread = get_strictness; optwrite = set_strictness } let tcl_change_info_gen info_gen = (fun gls -> let gl =sig_it gls in {it=[{gl with evar_extra=info_gen}];sigma=sig_sig gls}, function [pftree] -> {pftree with goal=gl; ref=Some (Prim Change_evars,[pftree])} | _ -> anomaly "change_info : Wrong number of subtrees") let tcl_change_info info gls = tcl_change_info_gen (Some (pm_in info)) gls let tcl_erase_info gls = tcl_change_info_gen None gls let special_whd gl= let infos=Closure.create_clos_infos Closure.betadeltaiota (pf_env gl) in (fun t -> Closure.whd_val infos (Closure.inject t)) let special_nf gl= let infos=Closure.create_clos_infos Closure.betaiotazeta (pf_env gl) in (fun t -> Closure.norm_val infos (Closure.inject t)) let is_good_inductive env ind = let mib,oib = Inductive.lookup_mind_specif env ind in oib.mind_nrealargs = 0 && not (Inductiveops.mis_is_recursive (ind,mib,oib)) let check_not_per pts = if not (Proof_trees.is_complete_proof (proof_of_pftreestate pts)) then match get_stack pts with Per (_,_,_,_)::_ -> error "You are inside a proof per cases/induction.\n\ Please \"suppose\" something or \"end\" it now." | _ -> () let mk_evd metalist gls = let evd0= create_goal_evar_defs (sig_sig gls) in let add_one (meta,typ) evd = meta_declare meta typ evd in List.fold_right add_one metalist evd0 let is_tmp id = (string_of_id id).[0] = '_' let tmp_ids gls = let ctx = pf_hyps gls in match ctx with [] -> [] | _::q -> List.filter is_tmp (ids_of_named_context q) let clean_tmp gls = let clean_id id0 gls0 = tclTRY (clear [id0]) gls0 in let rec clean_all = function [] -> tclIDTAC | id :: rest -> tclTHEN (clean_id id) (clean_all rest) in clean_all (tmp_ids gls) gls let assert_postpone id t = assert_tac (Name id) t (* start a proof *) let start_proof_tac gls= let gl=sig_it gls in let info={pm_stack=[]} in {it=[{gl with evar_extra=Some (pm_in info)}];sigma=sig_sig gls}, function [pftree] -> {pftree with goal=gl; ref=Some (Decl_proof true,[pftree])} | _ -> anomaly "Dem : Wrong number of subtrees" let go_to_proof_mode () = Pfedit.mutate (fun pts -> nth_unproven 1 (solve_pftreestate start_proof_tac pts)) (* closing gaps *) let daimon_tac gls = set_daimon_flag (); ({it=[];sigma=sig_sig gls}, function [] -> {open_subgoals=0; goal=sig_it gls; ref=Some (Daimon,[])} | _ -> anomaly "Daimon: Wrong number of subtrees") let daimon _ pftree = set_daimon_flag (); {pftree with open_subgoals=0; ref=Some (Daimon,[])} let daimon_subtree = map_pftreestate (fun _ -> frontier_mapi daimon ) (* marking closed blocks *) let rec is_focussing_instr = function Pthus i | Pthen i | Phence i -> is_focussing_instr i | Pescape | Pper _ | Pclaim _ | Pfocus _ | Psuppose _ | Pcase (_,_,_) -> true | _ -> false let mark_rule_as_done = function Decl_proof true -> Decl_proof false | Decl_proof false -> anomaly "already marked as done" | Nested(Proof_instr (lock_focus,instr),spfl) -> if lock_focus then Nested(Proof_instr (false,instr),spfl) else anomaly "already marked as done" | _ -> anomaly "mark_rule_as_done" let mark_proof_tree_as_done pt = match pt.ref with None -> anomaly "mark_proof_tree_as_done" | Some (r,spfl) -> {pt with ref= Some (mark_rule_as_done r,spfl)} let mark_as_done pts = map_pftreestate (fun _ -> mark_proof_tree_as_done) (up_to_matching_rule is_focussing_command pts) (* post-instruction focus management *) let goto_current_focus pts = up_until_matching_rule is_focussing_command pts let goto_current_focus_or_top pts = try up_until_matching_rule is_focussing_command pts with Not_found -> top_of_tree pts (* return *) let close_tactic_mode pts = let pts1= try goto_current_focus pts with Not_found -> error "\"return\" cannot be used outside of Declarative Proof Mode." in let pts2 = daimon_subtree pts1 in let pts3 = mark_as_done pts2 in goto_current_focus pts3 let return_from_tactic_mode () = Pfedit.mutate close_tactic_mode (* end proof/claim *) let close_block bt pts = let stack = if Proof_trees.is_complete_proof (proof_of_pftreestate pts) then get_top_stack pts else get_stack pts in match bt,stack with B_claim, Claim::_ | B_focus, Focus_claim::_ | B_proof, [] -> daimon_subtree (goto_current_focus pts) | _, Claim::_ -> error "\"end claim\" expected." | _, Focus_claim::_ -> error "\"end focus\" expected." | _, [] -> error "\"end proof\" expected." | _, (Per (et,_,_,_)::_|Suppose_case::Per (et,_,_,_)::_) -> begin match et with ET_Case_analysis -> error "\"end cases\" expected." | ET_Induction -> error "\"end induction\" expected." end | _,_ -> anomaly "Lonely suppose on stack." (* utility for suppose / suppose it is *) let close_previous_case pts = if Proof_trees.is_complete_proof (proof_of_pftreestate pts) then match get_top_stack pts with Per (et,_,_,_) :: _ -> anomaly "Weird case occured ..." | Suppose_case :: Per (et,_,_,_) :: _ -> goto_current_focus (mark_as_done pts) | _ -> error "Not inside a proof per cases or induction." else match get_stack pts with Per (et,_,_,_) :: _ -> pts | Suppose_case :: Per (et,_,_,_) :: _ -> goto_current_focus (mark_as_done (daimon_subtree pts)) | _ -> error "Not inside a proof per cases or induction." (* Proof instructions *) (* automation *) let filter_hyps f gls = let filter_aux (id,_,_) = if f id then tclIDTAC else tclTRY (clear [id]) in tclMAP filter_aux (Environ.named_context_of_val gls.it.evar_hyps) gls let local_hyp_prefix = id_of_string "___" let add_justification_hyps keep items gls = let add_aux c gls= match kind_of_term c with Var id -> keep:=Idset.add id !keep; tclIDTAC gls | _ -> let id=pf_get_new_id local_hyp_prefix gls in keep:=Idset.add id !keep; tclTHEN (letin_tac None (Names.Name id) c None Tacexpr.nowhere) (thin_body [id]) gls in tclMAP add_aux items gls let prepare_goal items gls = let tokeep = ref Idset.empty in let auxres = add_justification_hyps tokeep items gls in tclTHENLIST [ (fun _ -> auxres); filter_hyps (let keep = !tokeep in fun id -> Idset.mem id keep)] gls let my_automation_tac = ref (fun gls -> anomaly "No automation registered") let register_automation_tac tac = my_automation_tac:= tac let automation_tac gls = !my_automation_tac gls let justification tac gls= tclORELSE (tclSOLVE [tclTHEN tac assumption]) (fun gls -> if get_strictness () then error "Insufficient justification." else begin msg_warning (str "Insufficient justification."); daimon_tac gls end) gls let default_justification elems gls= justification (tclTHEN (prepare_goal elems) automation_tac) gls (* code for conclusion refining *) let constant dir s = lazy (Coqlib.gen_constant "Declarative" dir s) let _and = constant ["Init";"Logic"] "and" let _and_rect = constant ["Init";"Logic"] "and_rect" let _prod = constant ["Init";"Datatypes"] "prod" let _prod_rect = constant ["Init";"Datatypes"] "prod_rect" let _ex = constant ["Init";"Logic"] "ex" let _ex_ind = constant ["Init";"Logic"] "ex_ind" let _sig = constant ["Init";"Specif"] "sig" let _sig_rect = constant ["Init";"Specif"] "sig_rect" let _sigT = constant ["Init";"Specif"] "sigT" let _sigT_rect = constant ["Init";"Specif"] "sigT_rect" type stackd_elt = {se_meta:metavariable; se_type:types; se_last_meta:metavariable; se_meta_list:(metavariable*types) list; se_evd: evar_defs} let rec replace_in_list m l = function [] -> raise Not_found | c::q -> if m=fst c then l@q else c::replace_in_list m l q let enstack_subsubgoals env se stack gls= let hd,params = decompose_app (special_whd gls se.se_type) in match kind_of_term hd with Ind ind when is_good_inductive env ind -> let mib,oib= Inductive.lookup_mind_specif env ind in let gentypes= Inductive.arities_of_constructors ind (mib,oib) in let process i gentyp = let constructor = mkConstruct(ind,succ i) (* constructors numbering*) in let appterm = applist (constructor,params) in let apptype = Term.prod_applist gentyp params in let rc,_ = Reduction.dest_prod env apptype in let rec meta_aux last lenv = function [] -> (last,lenv,[]) | (nam,_,typ)::q -> let nlast=succ last in let (llast,holes,metas) = meta_aux nlast (mkMeta nlast :: lenv) q in (llast,holes,(nlast,special_nf gls (substl lenv typ))::metas) in let (nlast,holes,nmetas) = meta_aux se.se_last_meta [] (List.rev rc) in let refiner = applist (appterm,List.rev holes) in let evd = meta_assign se.se_meta (refiner,(ConvUpToEta 0,TypeProcessed (* ? *))) se.se_evd in let ncreated = replace_in_list se.se_meta nmetas se.se_meta_list in let evd0 = List.fold_left (fun evd (m,typ) -> meta_declare m typ evd) evd nmetas in List.iter (fun (m,typ) -> Stack.push {se_meta=m; se_type=typ; se_evd=evd0; se_meta_list=ncreated; se_last_meta=nlast} stack) (List.rev nmetas) in Array.iteri process gentypes | _ -> () let rec nf_list evd = function [] -> [] | (m,typ)::others -> if meta_defined evd m then nf_list evd others else (m,nf_meta evd typ)::nf_list evd others let find_subsubgoal c ctyp skip submetas gls = let env= pf_env gls in let concl = pf_concl gls in let evd = mk_evd ((0,concl)::submetas) gls in let stack = Stack.create () in let max_meta = List.fold_left (fun a (m,_) -> max a m) 0 submetas in let _ = Stack.push {se_meta=0; se_type=concl; se_last_meta=max_meta; se_meta_list=[0,concl]; se_evd=evd} stack in let rec dfs n = let se = Stack.pop stack in try let unifier = Unification.w_unify true env Reduction.CUMUL ctyp se.se_type se.se_evd in if n <= 0 then {se with se_evd=meta_assign se.se_meta (c,(ConvUpToEta 0,TypeNotProcessed (* ?? *))) unifier; se_meta_list=replace_in_list se.se_meta submetas se.se_meta_list} else dfs (pred n) with _ -> begin enstack_subsubgoals env se stack gls; dfs n end in let nse= try dfs skip with Stack.Empty -> raise Not_found in nf_list nse.se_evd nse.se_meta_list,nf_meta nse.se_evd (mkMeta 0) let concl_refiner metas body gls = let concl = pf_concl gls in let evd = sig_sig gls in let env = pf_env gls in let sort = family_of_sort (Typing.sort_of env evd concl) in let rec aux env avoid subst = function [] -> anomaly "concl_refiner: cannot happen" | (n,typ)::rest -> let _A = subst_meta subst typ in let x = id_of_name_using_hdchar env _A Anonymous in let _x = fresh_id avoid x gls in let nenv = Environ.push_named (_x,None,_A) env in let asort = family_of_sort (Typing.sort_of nenv evd _A) in let nsubst = (n,mkVar _x)::subst in if rest = [] then asort,_A,mkNamedLambda _x _A (subst_meta nsubst body) else let bsort,_B,nbody = aux nenv (_x::avoid) ((n,mkVar _x)::subst) rest in let body = mkNamedLambda _x _A nbody in if occur_term (mkVar _x) _B then begin let _P = mkNamedLambda _x _A _B in match bsort,sort with InProp,InProp -> let _AxB = mkApp(Lazy.force _ex,[|_A;_P|]) in InProp,_AxB, mkApp(Lazy.force _ex_ind,[|_A;_P;concl;body|]) | InProp,_ -> let _AxB = mkApp(Lazy.force _sig,[|_A;_P|]) in let _P0 = mkLambda(Anonymous,_AxB,concl) in InType,_AxB, mkApp(Lazy.force _sig_rect,[|_A;_P;_P0;body|]) | _,_ -> let _AxB = mkApp(Lazy.force _sigT,[|_A;_P|]) in let _P0 = mkLambda(Anonymous,_AxB,concl) in InType,_AxB, mkApp(Lazy.force _sigT_rect,[|_A;_P;_P0;body|]) end else begin match asort,bsort with InProp,InProp -> let _AxB = mkApp(Lazy.force _and,[|_A;_B|]) in InProp,_AxB, mkApp(Lazy.force _and_rect,[|_A;_B;concl;body|]) |_,_ -> let _AxB = mkApp(Lazy.force _prod,[|_A;_B|]) in let _P0 = mkLambda(Anonymous,_AxB,concl) in InType,_AxB, mkApp(Lazy.force _prod_rect,[|_A;_B;_P0;body|]) end in let (_,_,prf) = aux env [] [] metas in mkApp(prf,[|mkMeta 1|]) let thus_tac c ctyp submetas gls = let list,proof = try find_subsubgoal c ctyp 0 submetas gls with Not_found -> error "I could not relate this statement to the thesis." in if list = [] then exact_check proof gls else let refiner = concl_refiner list proof gls in Tactics.refine refiner gls (* general forward step *) let mk_stat_or_thesis info gls = function This c -> c | Thesis (For _ ) -> error "\"thesis for ...\" is not applicable here." | Thesis Plain -> pf_concl gls let just_tac _then cut info gls0 = let items_tac gls = match cut.cut_by with None -> tclIDTAC gls | Some items -> let items_ = if _then then let last_id = get_last (pf_env gls) in (mkVar last_id)::items else items in prepare_goal items_ gls in let method_tac gls = match cut.cut_using with None -> automation_tac gls | Some tac -> (Tacinterp.eval_tactic tac) gls in justification (tclTHEN items_tac method_tac) gls0 let instr_cut mkstat _thus _then cut gls0 = let info = get_its_info gls0 in let stat = cut.cut_stat in let (c_id,_) = match stat.st_label with Anonymous -> pf_get_new_id (id_of_string "_fact") gls0,false | Name id -> id,true in let c_stat = mkstat info gls0 stat.st_it in let thus_tac gls= if _thus then thus_tac (mkVar c_id) c_stat [] gls else tclIDTAC gls in tclTHENS (assert_postpone c_id c_stat) [tclTHEN tcl_erase_info (just_tac _then cut info); thus_tac] gls0 (* iterated equality *) let _eq = Libnames.constr_of_global (Coqlib.glob_eq) let decompose_eq id gls = let typ = pf_get_hyp_typ gls id in let whd = (special_whd gls typ) in match kind_of_term whd with App (f,args)-> if eq_constr f _eq && (Array.length args)=3 then (args.(0), args.(1), args.(2)) else error "Previous step is not an equality." | _ -> error "Previous step is not an equality." let instr_rew _thus rew_side cut gls0 = let last_id = try get_last (pf_env gls0) with _ -> error "No previous equality." in let typ,lhs,rhs = decompose_eq last_id gls0 in let items_tac gls = match cut.cut_by with None -> tclIDTAC gls | Some items -> prepare_goal items gls in let method_tac gls = match cut.cut_using with None -> automation_tac gls | Some tac -> (Tacinterp.eval_tactic tac) gls in let just_tac gls = justification (tclTHEN items_tac method_tac) gls in let (c_id,_) = match cut.cut_stat.st_label with Anonymous -> pf_get_new_id (id_of_string "_eq") gls0,false | Name id -> id,true in let thus_tac new_eq gls= if _thus then thus_tac (mkVar c_id) new_eq [] gls else tclIDTAC gls in match rew_side with Lhs -> let new_eq = mkApp(_eq,[|typ;cut.cut_stat.st_it;rhs|]) in tclTHENS (assert_postpone c_id new_eq) [tclTHEN tcl_erase_info (tclTHENS (transitivity lhs) [just_tac;exact_check (mkVar last_id)]); thus_tac new_eq] gls0 | Rhs -> let new_eq = mkApp(_eq,[|typ;lhs;cut.cut_stat.st_it|]) in tclTHENS (assert_postpone c_id new_eq) [tclTHEN tcl_erase_info (tclTHENS (transitivity rhs) [exact_check (mkVar last_id);just_tac]); thus_tac new_eq] gls0 (* tactics for claim/focus *) let instr_claim _thus st gls0 = let info = get_its_info gls0 in let (id,_) = match st.st_label with Anonymous -> pf_get_new_id (id_of_string "_claim") gls0,false | Name id -> id,true in let thus_tac gls= if _thus then thus_tac (mkVar id) st.st_it [] gls else tclIDTAC gls in let ninfo1 = {pm_stack= (if _thus then Focus_claim else Claim)::info.pm_stack} in tclTHENS (assert_postpone id st.st_it) [tcl_change_info ninfo1; thus_tac] gls0 (* tactics for assume *) let push_intro_tac coerce nam gls = let (hid,_) = match nam with Anonymous -> pf_get_new_id (id_of_string "_hyp") gls,false | Name id -> id,true in tclTHENLIST [intro_mustbe_force hid; coerce hid] gls let assume_tac hyps gls = List.fold_right (fun (Hvar st | Hprop st) -> tclTHEN (push_intro_tac (fun id -> convert_hyp (id,None,st.st_it)) st.st_label)) hyps tclIDTAC gls let assume_hyps_or_theses hyps gls = List.fold_right (function (Hvar {st_label=nam;st_it=c} | Hprop {st_label=nam;st_it=This c}) -> tclTHEN (push_intro_tac (fun id -> convert_hyp (id,None,c)) nam) | Hprop {st_label=nam;st_it=Thesis (tk)} -> tclTHEN (push_intro_tac (fun id -> tclIDTAC) nam)) hyps tclIDTAC gls let assume_st hyps gls = List.fold_right (fun st -> tclTHEN (push_intro_tac (fun id -> convert_hyp (id,None,st.st_it)) st.st_label)) hyps tclIDTAC gls let assume_st_letin hyps gls = List.fold_right (fun st -> tclTHEN (push_intro_tac (fun id -> convert_hyp (id,Some (fst st.st_it),snd st.st_it)) st.st_label)) hyps tclIDTAC gls (* suffices *) let rec metas_from n hyps = match hyps with _ :: q -> n :: metas_from (succ n) q | [] -> [] let rec build_product args body = match args with (Hprop st| Hvar st )::rest -> let pprod= lift 1 (build_product rest body) in let lbody = match st.st_label with Anonymous -> pprod | Name id -> subst_term (mkVar id) pprod in mkProd (st.st_label, st.st_it, lbody) | [] -> body let rec build_applist prod = function [] -> [],prod | n::q -> let (_,typ,_) = destProd prod in let ctx,head = build_applist (Term.prod_applist prod [mkMeta n]) q in (n,typ)::ctx,head let instr_suffices _then cut gls0 = let info = get_its_info gls0 in let c_id = pf_get_new_id (id_of_string "_cofact") gls0 in let ctx,hd = cut.cut_stat in let c_stat = build_product ctx (mk_stat_or_thesis info gls0 hd) in let metas = metas_from 1 ctx in let c_ctx,c_head = build_applist c_stat metas in let c_term = applist (mkVar c_id,List.map mkMeta metas) in let thus_tac gls= thus_tac c_term c_head c_ctx gls in tclTHENS (assert_postpone c_id c_stat) [tclTHENLIST [ assume_tac ctx; tcl_erase_info; just_tac _then cut info]; thus_tac] gls0 (* tactics for consider/given *) let conjunction_arity id gls = let typ = pf_get_hyp_typ gls id in let hd,params = decompose_app (special_whd gls typ) in let env =pf_env gls in match kind_of_term hd with Ind ind when is_good_inductive env ind -> let mib,oib= Inductive.lookup_mind_specif env ind in let gentypes= Inductive.arities_of_constructors ind (mib,oib) in let _ = if Array.length gentypes <> 1 then raise Not_found in let apptype = Term.prod_applist gentypes.(0) params in let rc,_ = Reduction.dest_prod env apptype in List.length rc | _ -> raise Not_found let rec intron_then n ids ltac gls = if n<=0 then ltac ids gls else let id = pf_get_new_id (id_of_string "_tmp") gls in tclTHEN (intro_mustbe_force id) (intron_then (pred n) (id::ids) ltac) gls let rec consider_match may_intro introduced available expected gls = match available,expected with [],[] -> tclIDTAC gls | _,[] -> error "Last statements do not match a complete hypothesis." (* should tell which ones *) | [],hyps -> if may_intro then begin let id = pf_get_new_id (id_of_string "_tmp") gls in tclIFTHENELSE (intro_mustbe_force id) (consider_match true [] [id] hyps) (fun _ -> error "Not enough sub-hypotheses to match statements.") gls end else error "Not enough sub-hypotheses to match statements." (* should tell which ones *) | id::rest_ids,(Hvar st | Hprop st)::rest -> tclIFTHENELSE (convert_hyp (id,None,st.st_it)) begin match st.st_label with Anonymous -> consider_match may_intro ((id,false)::introduced) rest_ids rest | Name hid -> tclTHENLIST [rename_hyp [id,hid]; consider_match may_intro ((hid,true)::introduced) rest_ids rest] end begin (fun gls -> let nhyps = try conjunction_arity id gls with Not_found -> error "Matching hypothesis not found." in tclTHENLIST [general_case_analysis false (mkVar id,NoBindings); intron_then nhyps [] (fun l -> consider_match may_intro introduced (List.rev_append l rest_ids) expected)] gls) end gls let consider_tac c hyps gls = match kind_of_term (strip_outer_cast c) with Var id -> consider_match false [] [id] hyps gls | _ -> let id = pf_get_new_id (id_of_string "_tmp") gls in tclTHEN (forward None (Some (dummy_loc, Genarg.IntroIdentifier id)) c) (consider_match false [] [id] hyps) gls let given_tac hyps gls = consider_match true [] [] hyps gls (* tactics for take *) let rec take_tac wits gls = match wits with [] -> tclIDTAC gls | wit::rest -> let typ = pf_type_of gls wit in tclTHEN (thus_tac wit typ []) (take_tac rest) gls (* tactics for define *) let rec build_function args body = match args with st::rest -> let pfun= lift 1 (build_function rest body) in let id = match st.st_label with Anonymous -> assert false | Name id -> id in mkLambda (Name id, st.st_it, subst_term (mkVar id) pfun) | [] -> body let define_tac id args body gls = let t = build_function args body in letin_tac None (Name id) t None Tacexpr.nowhere gls (* tactics for reconsider *) let cast_tac id_or_thesis typ gls = match id_or_thesis with This id -> let (_,body,_) = pf_get_hyp gls id in convert_hyp (id,body,typ) gls | Thesis (For _ ) -> error "\"thesis for ...\" is not applicable here." | Thesis Plain -> convert_concl typ DEFAULTcast gls (* per cases *) let is_rec_pos (main_ind,wft) = match main_ind with None -> false | Some index -> match fst (Rtree.dest_node wft) with Mrec i when i = index -> true | _ -> false let rec constr_trees (main_ind,wft) ind = match Rtree.dest_node wft with Norec,_ -> let itree = (snd (Global.lookup_inductive ind)).mind_recargs in constr_trees (None,itree) ind | _,constrs -> main_ind,constrs let ind_args rp ind = let main_ind,constrs = constr_trees rp ind in let args ctree = Array.map (fun t -> main_ind,t) (snd (Rtree.dest_node ctree)) in Array.map args constrs let init_tree ids ind rp nexti = let indargs = ind_args rp ind in let do_i i arp = (Array.map is_rec_pos arp),nexti i arp in Split_patt (ids,ind,Array.mapi do_i indargs) let map_tree_rp rp id_fun mapi = function Split_patt (ids,ind,branches) -> let indargs = ind_args rp ind in let do_i i (recargs,bri) = recargs,mapi i indargs.(i) bri in Split_patt (id_fun ids,ind,Array.mapi do_i branches) | _ -> failwith "map_tree_rp: not a splitting node" let map_tree id_fun mapi = function Split_patt (ids,ind,branches) -> let do_i i (recargs,bri) = recargs,mapi i bri in Split_patt (id_fun ids,ind,Array.mapi do_i branches) | _ -> failwith "map_tree: not a splitting node" let start_tree env ind rp = init_tree Idset.empty ind rp (fun _ _ -> None) let build_per_info etype casee gls = let concl=pf_concl gls in let env=pf_env gls in let ctyp=pf_type_of gls casee in let is_dep = dependent casee concl in let hd,args = decompose_app (special_whd gls ctyp) in let ind = try destInd hd with _ -> error "Case analysis must be done on an inductive object." in let mind,oind = Global.lookup_inductive ind in let nparams,index = match etype with ET_Induction -> mind.mind_nparams_rec,Some (snd ind) | _ -> mind.mind_nparams,None in let params,real_args = list_chop nparams args in let abstract_obj c body = let typ=pf_type_of gls c in lambda_create env (typ,subst_term c body) in let pred= List.fold_right abstract_obj real_args (lambda_create env (ctyp,subst_term casee concl)) in is_dep, {per_casee=casee; per_ctype=ctyp; per_ind=ind; per_pred=pred; per_args=real_args; per_params=params; per_nparams=nparams; per_wf=index,oind.mind_recargs} let per_tac etype casee gls= let env=pf_env gls in let info = get_its_info gls in match casee with Real c -> let is_dep,per_info = build_per_info etype c gls in let ek = if is_dep then EK_dep (start_tree env per_info.per_ind per_info.per_wf) else EK_unknown in tcl_change_info {pm_stack= Per(etype,per_info,ek,[])::info.pm_stack} gls | Virtual cut -> assert (cut.cut_stat.st_label=Anonymous); let id = pf_get_new_id (id_of_string "anonymous_matched") gls in let c = mkVar id in let modified_cut = {cut with cut_stat={cut.cut_stat with st_label=Name id}} in tclTHEN (instr_cut (fun _ _ c -> c) false false modified_cut) (fun gls0 -> let is_dep,per_info = build_per_info etype c gls0 in assert (not is_dep); tcl_change_info {pm_stack= Per(etype,per_info,EK_unknown,[])::info.pm_stack} gls0) gls (* suppose *) let register_nodep_subcase id= function Per(et,pi,ek,clauses)::s -> begin match ek with EK_unknown -> clauses,Per(et,pi,EK_nodep,id::clauses)::s | EK_nodep -> clauses,Per(et,pi,EK_nodep,id::clauses)::s | EK_dep _ -> error "Do not mix \"suppose\" with \"suppose it is\"." end | _ -> anomaly "wrong stack state" let suppose_tac hyps gls0 = let info = get_its_info gls0 in let thesis = pf_concl gls0 in let id = pf_get_new_id (id_of_string "subcase_") gls0 in let clause = build_product hyps thesis in let ninfo1 = {pm_stack=Suppose_case::info.pm_stack} in let old_clauses,stack = register_nodep_subcase id info.pm_stack in let ninfo2 = {pm_stack=stack} in tclTHENS (assert_postpone id clause) [tclTHENLIST [tcl_change_info ninfo1; assume_tac hyps; clear old_clauses]; tcl_change_info ninfo2] gls0 (* suppose it is ... *) (* pattern matching compiling *) let rec skip_args rest ids n = if n <= 0 then Close_patt rest else Skip_patt (ids,skip_args rest ids (pred n)) let rec tree_of_pats ((id,_) as cpl) pats = match pats with [] -> End_patt cpl | args::stack -> match args with [] -> Close_patt (tree_of_pats cpl stack) | (patt,rp) :: rest_args -> match patt with PatVar (_,v) -> Skip_patt (Idset.singleton id, tree_of_pats cpl (rest_args::stack)) | PatCstr (_,(ind,cnum),args,nam) -> let nexti i ati = if i = pred cnum then let nargs = list_map_i (fun j a -> (a,ati.(j))) 0 args in Some (Idset.singleton id, tree_of_pats cpl (nargs::rest_args::stack)) else None in init_tree Idset.empty ind rp nexti let rec add_branch ((id,_) as cpl) pats tree= match pats with [] -> begin match tree with End_patt cpl0 -> End_patt cpl0 (* this ensures precedence for overlapping patterns *) | _ -> anomaly "tree is expected to end here" end | args::stack -> match args with [] -> begin match tree with Close_patt t -> Close_patt (add_branch cpl stack t) | _ -> anomaly "we should pop here" end | (patt,rp) :: rest_args -> match patt with PatVar (_,v) -> begin match tree with Skip_patt (ids,t) -> Skip_patt (Idset.add id ids, add_branch cpl (rest_args::stack) t) | Split_patt (_,_,_) -> map_tree (Idset.add id) (fun i bri -> append_branch cpl 1 (rest_args::stack) bri) tree | _ -> anomaly "No pop/stop expected here" end | PatCstr (_,(ind,cnum),args,nam) -> match tree with Skip_patt (ids,t) -> let nexti i ati = if i = pred cnum then let nargs = list_map_i (fun j a -> (a,ati.(j))) 0 args in Some (Idset.add id ids, add_branch cpl (nargs::rest_args::stack) (skip_args t ids (Array.length ati))) else Some (ids, skip_args t ids (Array.length ati)) in init_tree ids ind rp nexti | Split_patt (_,ind0,_) -> if (ind <> ind0) then error (* this can happen with coercions *) "Case pattern belongs to wrong inductive type."; let mapi i ati bri = if i = pred cnum then let nargs = list_map_i (fun j a -> (a,ati.(j))) 0 args in append_branch cpl 0 (nargs::rest_args::stack) bri else bri in map_tree_rp rp (fun ids -> ids) mapi tree | _ -> anomaly "No pop/stop expected here" and append_branch ((id,_) as cpl) depth pats = function Some (ids,tree) -> Some (Idset.add id ids,append_tree cpl depth pats tree) | None -> Some (Idset.singleton id,tree_of_pats cpl pats) and append_tree ((id,_) as cpl) depth pats tree = if depth<=0 then add_branch cpl pats tree else match tree with Close_patt t -> Close_patt (append_tree cpl (pred depth) pats t) | Skip_patt (ids,t) -> Skip_patt (Idset.add id ids,append_tree cpl depth pats t) | End_patt _ -> anomaly "Premature end of branch" | Split_patt (_,_,_) -> map_tree (Idset.add id) (fun i bri -> append_branch cpl (succ depth) pats bri) tree (* suppose it is *) let rec st_assoc id = function [] -> raise Not_found | st::_ when st.st_label = id -> st.st_it | _ :: rest -> st_assoc id rest let thesis_for obj typ per_info env= let rc,hd1=decompose_prod typ in let cind,all_args=decompose_app typ in let ind = destInd cind in let _ = if ind <> per_info.per_ind then errorlabstrm "thesis_for" ((Printer.pr_constr_env env obj) ++ spc () ++ str"cannot give an induction hypothesis (wrong inductive type).") in let params,args = list_chop per_info.per_nparams all_args in let _ = if not (List.for_all2 eq_constr params per_info.per_params) then errorlabstrm "thesis_for" ((Printer.pr_constr_env env obj) ++ spc () ++ str "cannot give an induction hypothesis (wrong parameters).") in let hd2 = (applist ((lift (List.length rc) per_info.per_pred),args@[obj])) in compose_prod rc (whd_beta Evd.empty hd2) let rec build_product_dep pat_info per_info args body gls = match args with (Hprop {st_label=nam;st_it=This c} | Hvar {st_label=nam;st_it=c})::rest -> let pprod= lift 1 (build_product_dep pat_info per_info rest body gls) in let lbody = match nam with Anonymous -> body | Name id -> subst_var id pprod in mkProd (nam,c,lbody) | Hprop ({st_it=Thesis tk} as st)::rest -> let pprod= lift 1 (build_product_dep pat_info per_info rest body gls) in let lbody = match st.st_label with Anonymous -> body | Name id -> subst_var id pprod in let ptyp = match tk with For id -> let obj = mkVar id in let typ = try st_assoc (Name id) pat_info.pat_vars with Not_found -> snd (st_assoc (Name id) pat_info.pat_aliases) in thesis_for obj typ per_info (pf_env gls) | Plain -> pf_concl gls in mkProd (st.st_label,ptyp,lbody) | [] -> body let build_dep_clause params pat_info per_info hyps gls = let concl= thesis_for pat_info.pat_constr pat_info.pat_typ per_info (pf_env gls) in let open_clause = build_product_dep pat_info per_info hyps concl gls in let prod_one st body = match st.st_label with Anonymous -> mkProd(Anonymous,st.st_it,lift 1 body) | Name id -> mkNamedProd id st.st_it (lift 1 body) in let let_one_in st body = match st.st_label with Anonymous -> mkLetIn(Anonymous,fst st.st_it,snd st.st_it,lift 1 body) | Name id -> mkNamedLetIn id (fst st.st_it) (snd st.st_it) (lift 1 body) in let aliased_clause = List.fold_right let_one_in pat_info.pat_aliases open_clause in List.fold_right prod_one (params@pat_info.pat_vars) aliased_clause let rec register_dep_subcase id env per_info pat = function EK_nodep -> error "Only \"suppose it is\" can be used here." | EK_unknown -> register_dep_subcase id env per_info pat (EK_dep (start_tree env per_info.per_ind per_info.per_wf)) | EK_dep tree -> EK_dep (add_branch id [[pat,per_info.per_wf]] tree) let case_tac params pat_info hyps gls0 = let info = get_its_info gls0 in let id = pf_get_new_id (id_of_string "subcase_") gls0 in let et,per_info,ek,old_clauses,rest = match info.pm_stack with Per (et,pi,ek,old_clauses)::rest -> (et,pi,ek,old_clauses,rest) | _ -> anomaly "wrong place for cases" in let clause = build_dep_clause params pat_info per_info hyps gls0 in let ninfo1 = {pm_stack=Suppose_case::info.pm_stack} in let nek = register_dep_subcase (id,List.length hyps) (pf_env gls0) per_info pat_info.pat_pat ek in let ninfo2 = {pm_stack=Per(et,per_info,nek,id::old_clauses)::rest} in tclTHENS (assert_postpone id clause) [tclTHENLIST [tcl_change_info ninfo1; assume_st (params@pat_info.pat_vars); assume_st_letin pat_info.pat_aliases; assume_hyps_or_theses hyps; clear old_clauses]; tcl_change_info ninfo2] gls0 (* end cases *) type instance_stack = (constr option*(constr list) list) list let initial_instance_stack ids = List.map (fun id -> id,[None,[]]) ids let push_one_arg arg = function [] -> anomaly "impossible" | (head,args) :: ctx -> ((head,(arg::args)) :: ctx) let push_arg arg stacks = List.map (fun (id,stack) -> (id,push_one_arg arg stack)) stacks let push_one_head c ids (id,stack) = let head = if Idset.mem id ids then Some c else None in id,(head,[]) :: stack let push_head c ids stacks = List.map (push_one_head c ids) stacks let pop_one (id,stack) = let nstack= match stack with [] -> anomaly "impossible" | [c] as l -> l | (Some head,args)::(head0,args0)::ctx -> let arg = applist (head,(List.rev args)) in (head0,(arg::args0))::ctx | (None,args)::(head0,args0)::ctx -> (head0,(args@args0))::ctx in id,nstack let pop_stacks stacks = List.map pop_one stacks let hrec_for fix_id per_info gls obj_id = let obj=mkVar obj_id in let typ=pf_get_hyp_typ gls obj_id in let rc,hd1=decompose_prod typ in let cind,all_args=decompose_app typ in let ind = destInd cind in assert (ind=per_info.per_ind); let params,args= list_chop per_info.per_nparams all_args in assert begin try List.for_all2 eq_constr params per_info.per_params with Invalid_argument _ -> false end; let hd2 = applist (mkVar fix_id,args@[obj]) in compose_lam rc (whd_beta gls.sigma hd2) let rec execute_cases fix_name per_info tacnext args objs nhrec tree gls = match tree, objs with Close_patt t,_ -> let args0 = pop_stacks args in execute_cases fix_name per_info tacnext args0 objs nhrec t gls | Skip_patt (_,t),skipped::next_objs -> let args0 = push_arg skipped args in execute_cases fix_name per_info tacnext args0 next_objs nhrec t gls | End_patt (id,nhyps),[] -> begin match List.assoc id args with [None,br_args] -> let metas = list_tabulate (fun n -> mkMeta (succ n)) nhyps in tclTHEN (tclDO nhrec introf) (tacnext (applist (mkVar id,List.rev_append br_args metas))) gls | _ -> anomaly "wrong stack size" end | Split_patt (ids,ind,br), casee::next_objs -> let (mind,oind) as spec = Global.lookup_inductive ind in let nparams = mind.mind_nparams in let concl=pf_concl gls in let env=pf_env gls in let ctyp=pf_type_of gls casee in let hd,all_args = decompose_app (special_whd gls ctyp) in let _ = assert (destInd hd = ind) in (* just in case *) let params,real_args = list_chop nparams all_args in let abstract_obj c body = let typ=pf_type_of gls c in lambda_create env (typ,subst_term c body) in let elim_pred = List.fold_right abstract_obj real_args (lambda_create env (ctyp,subst_term casee concl)) in let case_info = Inductiveops.make_case_info env ind RegularStyle in let gen_arities = Inductive.arities_of_constructors ind spec in let f_ids typ = let sign = (prod_assum (Term.prod_applist typ params)) in find_intro_names sign gls in let constr_args_ids = Array.map f_ids gen_arities in let case_term = mkCase(case_info,elim_pred,casee, Array.mapi (fun i _ -> mkMeta (succ i)) constr_args_ids) in let branch_tac i (recargs,bro) gls0 = let args_ids = constr_args_ids.(i) in let rec aux n = function [] -> assert (n=Array.length recargs); next_objs,[],nhrec | id :: q -> let objs,recs,nrec = aux (succ n) q in if recargs.(n) then (mkVar id::objs),(id::recs),succ nrec else (mkVar id::objs),recs,nrec in let objs,recs,nhrec = aux 0 args_ids in tclTHENLIST [tclMAP intro_mustbe_force args_ids; begin fun gls1 -> let hrecs = List.map (fun id -> hrec_for (out_name fix_name) per_info gls1 id) recs in generalize hrecs gls1 end; match bro with None -> msg_warning (str "missing case"); tacnext (mkMeta 1) | Some (sub_ids,tree) -> let br_args = List.filter (fun (id,_) -> Idset.mem id sub_ids) args in let construct = applist (mkConstruct(ind,succ i),params) in let p_args = push_head construct ids br_args in execute_cases fix_name per_info tacnext p_args objs nhrec tree] gls0 in tclTHENSV (refine case_term) (Array.mapi branch_tac br) gls | Split_patt (_, _, _) , [] -> anomaly "execute_cases : Nothing to split" | Skip_patt _ , [] -> anomaly "execute_cases : Nothing to skip" | End_patt (_,_) , _ :: _ -> anomaly "execute_cases : End of branch with garbage left" (* end focus/claim *) let end_tac et2 gls = let info = get_its_info gls in let et1,pi,ek,clauses = match info.pm_stack with Suppose_case::_ -> anomaly "This case should already be trapped" | Claim::_ -> error "\"end claim\" expected." | Focus_claim::_ -> error "\"end focus\" expected." | Per(et',pi,ek,clauses)::_ -> (et',pi,ek,clauses) | [] -> anomaly "This case should already be trapped" in let et = if et1 <> et2 then match et1 with ET_Case_analysis -> error "\"end cases\" expected." | ET_Induction -> error "\"end induction\" expected." else et1 in tclTHEN tcl_erase_info begin match et,ek with _,EK_unknown -> tclSOLVE [simplest_elim pi.per_casee] | ET_Case_analysis,EK_nodep -> tclTHEN (general_case_analysis false (pi.per_casee,NoBindings)) (default_justification (List.map mkVar clauses)) | ET_Induction,EK_nodep -> tclTHENLIST [generalize (pi.per_args@[pi.per_casee]); simple_induct (AnonHyp (succ (List.length pi.per_args))); default_justification (List.map mkVar clauses)] | ET_Case_analysis,EK_dep tree -> execute_cases Anonymous pi (fun c -> tclTHENLIST [refine c; clear clauses; justification assumption]) (initial_instance_stack clauses) [pi.per_casee] 0 tree | ET_Induction,EK_dep tree -> let nargs = (List.length pi.per_args) in tclTHEN (generalize (pi.per_args@[pi.per_casee])) begin fun gls0 -> let fix_id = pf_get_new_id (id_of_string "_fix") gls0 in let c_id = pf_get_new_id (id_of_string "_main_arg") gls0 in tclTHENLIST [fix (Some fix_id) (succ nargs); tclDO nargs introf; intro_mustbe_force c_id; execute_cases (Name fix_id) pi (fun c -> tclTHENLIST [clear [fix_id]; refine c; clear clauses; justification assumption]) (initial_instance_stack clauses) [mkVar c_id] 0 tree] gls0 end end gls (* escape *) let escape_tac gls = tcl_erase_info gls (* General instruction engine *) let rec do_proof_instr_gen _thus _then instr = match instr with Pthus i -> assert (not _thus); do_proof_instr_gen true _then i | Pthen i -> assert (not _then); do_proof_instr_gen _thus true i | Phence i -> assert (not (_then || _thus)); do_proof_instr_gen true true i | Pcut c -> instr_cut mk_stat_or_thesis _thus _then c | Psuffices c -> instr_suffices _then c | Prew (s,c) -> assert (not _then); instr_rew _thus s c | Pconsider (c,hyps) -> consider_tac c hyps | Pgiven hyps -> given_tac hyps | Passume hyps -> assume_tac hyps | Plet hyps -> assume_tac hyps | Pclaim st -> instr_claim false st | Pfocus st -> instr_claim true st | Ptake witl -> take_tac witl | Pdefine (id,args,body) -> define_tac id args body | Pcast (id,typ) -> cast_tac id typ | Pper (et,cs) -> per_tac et cs | Psuppose hyps -> suppose_tac hyps | Pcase (params,pat_info,hyps) -> case_tac params pat_info hyps | Pend (B_elim et) -> end_tac et | Pend _ -> anomaly "Not applicable" | Pescape -> escape_tac let eval_instr {instr=instr} = do_proof_instr_gen false false instr let rec preprocess pts instr = match instr with Phence i |Pthus i | Pthen i -> preprocess pts i | Psuffices _ | Pcut _ | Passume _ | Plet _ | Pclaim _ | Pfocus _ | Pconsider (_,_) | Pcast (_,_) | Pgiven _ | Ptake _ | Pdefine (_,_,_) | Pper _ | Prew _ -> check_not_per pts; true,pts | Pescape -> check_not_per pts; true,pts | Pcase _ | Psuppose _ | Pend (B_elim _) -> true,close_previous_case pts | Pend bt -> false,close_block bt pts let rec postprocess pts instr = match instr with Phence i | Pthus i | Pthen i -> postprocess pts i | Pcut _ | Psuffices _ | Passume _ | Plet _ | Pconsider (_,_) | Pcast (_,_) | Pgiven _ | Ptake _ | Pdefine (_,_,_) | Prew (_,_) -> pts | Pclaim _ | Pfocus _ | Psuppose _ | Pcase _ | Pper _ | Pescape -> nth_unproven 1 pts | Pend (B_elim ET_Induction) -> begin let pf = proof_of_pftreestate pts in let (pfterm,_) = extract_open_pftreestate pts in let env = Evd.evar_env (goal_of_proof pf) in try Inductiveops.control_only_guard env pfterm; goto_current_focus_or_top (mark_as_done pts) with Type_errors.TypeError(env, Type_errors.IllFormedRecBody(_,_,_,_,_)) -> anomaly "\"end induction\" generated an ill-formed fixpoint" end | Pend _ -> goto_current_focus_or_top (mark_as_done pts) let do_instr raw_instr pts = let has_tactic,pts1 = preprocess pts raw_instr.instr in let pts2 = if has_tactic then let gl = nth_goal_of_pftreestate 1 pts1 in let env= pf_env gl in let sigma= project gl in let ist = {ltacvars = ([],[]); ltacrecvars = []; gsigma = sigma; genv = env} in let glob_instr = intern_proof_instr ist raw_instr in let instr = interp_proof_instr (get_its_info gl) sigma env glob_instr in let lock_focus = is_focussing_instr instr.instr in let marker= Proof_instr (lock_focus,instr) in solve_nth_pftreestate 1 (abstract_operation marker (tclTHEN (eval_instr instr) clean_tmp)) pts1 else pts1 in postprocess pts2 raw_instr.instr let proof_instr raw_instr = Pfedit.mutate (do_instr raw_instr) (* (* STUFF FOR ITERATED RELATIONS *) let decompose_bin_app t= let hd,args = destApp let identify_transitivity_lemma c = let varx,tx,c1 = destProd c in let vary,ty,c2 = destProd (pop c1) in let varz,tz,c3 = destProd (pop c2) in let _,p1,c4 = destProd (pop c3) in let _,lp2,lp3 = destProd (pop c4) in let p2=pop lp2 in let p3=pop lp3 in *)