(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* logical_kind -> constr -> global_reference) = fun f_id kind value -> let ce = {const_entry_body = value; const_entry_secctx = None; const_entry_type = None; const_entry_opaque = false } in ConstRef(declare_constant f_id (DefinitionEntry ce, kind));; let defined () = Lemmas.save_named false let def_of_const t = match (kind_of_term t) with Const sp -> (try (match body_of_constant (Global.lookup_constant sp) with | Some c -> Declarations.force c | _ -> assert false) with _ -> anomaly ("Cannot find definition of constant "^ (string_of_id (id_of_label (con_label sp)))) ) |_ -> assert false let type_of_const t = match (kind_of_term t) with Const sp -> Typeops.type_of_constant (Global.env()) sp |_ -> assert false let constant sl s = constr_of_global (locate (make_qualid(Names.make_dirpath (List.map id_of_string (List.rev sl))) (id_of_string s)));; let const_of_ref = function ConstRef kn -> kn | _ -> anomaly "ConstRef expected" let nf_zeta env = Reductionops.clos_norm_flags (Closure.RedFlags.mkflags [Closure.RedFlags.fZETA]) env Evd.empty let nf_betaiotazeta = (* Reductionops.local_strong Reductionops.whd_betaiotazeta *) let clos_norm_flags flgs env sigma t = Closure.norm_val (Closure.create_clos_infos flgs env) (Closure.inject (Reductionops.nf_evar sigma t)) in clos_norm_flags Closure.betaiotazeta Environ.empty_env Evd.empty (* Generic values *) let pf_get_new_ids idl g = let ids = pf_ids_of_hyps g in List.fold_right (fun id acc -> next_global_ident_away id (acc@ids)::acc) idl [] let compute_renamed_type gls c = rename_bound_vars_as_displayed (*no avoid*) [] (*no rels*) [] (pf_type_of gls c) let h'_id = id_of_string "h'" let heq_id = id_of_string "Heq" let teq_id = id_of_string "teq" let ano_id = id_of_string "anonymous" let x_id = id_of_string "x" let k_id = id_of_string "k" let v_id = id_of_string "v" let def_id = id_of_string "def" let p_id = id_of_string "p" let rec_res_id = id_of_string "rec_res";; let lt = function () -> (coq_base_constant "lt") let le = function () -> (coq_base_constant "le") let ex = function () -> (coq_base_constant "ex") let nat = function () -> (coq_base_constant "nat") let coq_sig = function () -> (coq_base_constant "sig") let iter_ref () = try find_reference ["Recdef"] "iter" with Not_found -> error "module Recdef not loaded" let iter = function () -> (constr_of_global (delayed_force iter_ref)) let eq = function () -> (coq_base_constant "eq") let le_lt_SS = function () -> (constant ["Recdef"] "le_lt_SS") let le_lt_n_Sm = function () -> (coq_base_constant "le_lt_n_Sm") let le_trans = function () -> (coq_base_constant "le_trans") let le_lt_trans = function () -> (coq_base_constant "le_lt_trans") let lt_S_n = function () -> (coq_base_constant "lt_S_n") let le_n = function () -> (coq_base_constant "le_n") let coq_sig_ref = function () -> (find_reference ["Coq";"Init";"Specif"] "sig") let coq_O = function () -> (coq_base_constant "O") let coq_S = function () -> (coq_base_constant "S") let gt_antirefl = function () -> (coq_constant "gt_irrefl") let lt_n_O = function () -> (coq_base_constant "lt_n_O") let lt_n_Sn = function () -> (coq_base_constant "lt_n_Sn") let f_equal = function () -> (coq_constant "f_equal") let well_founded_induction = function () -> (coq_constant "well_founded_induction") let max_ref = function () -> (find_reference ["Recdef"] "max") let max_constr = function () -> (constr_of_global (delayed_force max_ref)) let coq_conj = function () -> find_reference ["Coq";"Init";"Logic"] "conj" let f_S t = mkApp(delayed_force coq_S, [|t|]);; let make_refl_eq constructor type_of_t t = mkApp(constructor,[|type_of_t;t|]) let rec n_x_id ids n = if n = 0 then [] else let x = next_ident_away_in_goal x_id ids in x::n_x_id (x::ids) (n-1);; let simpl_iter clause = reduce (Lazy {rBeta=true;rIota=true;rZeta= true; rDelta=false; rConst = [ EvalConstRef (const_of_ref (delayed_force iter_ref))]}) clause (* Others ugly things ... *) let (value_f:constr list -> global_reference -> constr) = fun al fterm -> let d0 = dummy_loc in let rev_x_id_l = ( List.fold_left (fun x_id_l _ -> let x_id = next_ident_away_in_goal x_id x_id_l in x_id::x_id_l ) [] al ) in let context = List.map (fun (x, c) -> Name x, None, c) (List.combine rev_x_id_l (List.rev al)) in let env = Environ.push_rel_context context (Global.env ()) in let glob_body = GCases (d0,RegularStyle,None, [GApp(d0, GRef(d0,fterm), List.rev_map (fun x_id -> GVar(d0, x_id)) rev_x_id_l), (Anonymous,None)], [d0, [v_id], [PatCstr(d0,(destIndRef (delayed_force coq_sig_ref),1), [PatVar(d0, Name v_id); PatVar(d0, Anonymous)], Anonymous)], GVar(d0,v_id)]) in let body = understand Evd.empty env glob_body in it_mkLambda_or_LetIn body context let (declare_f : identifier -> logical_kind -> constr list -> global_reference -> global_reference) = fun f_id kind input_type fterm_ref -> declare_fun f_id kind (value_f input_type fterm_ref);; (* Debuging mechanism *) let debug_queue = Stack.create () let rec print_debug_queue b e = if not (Stack.is_empty debug_queue) then begin let lmsg,goal = Stack.pop debug_queue in if b then Pp.msgnl (lmsg ++ (str " raised exception " ++ Errors.print e) ++ str " on goal " ++ goal) else begin Pp.msgnl (str " from " ++ lmsg ++ str " on goal " ++ goal); end; print_debug_queue false e; end let observe strm = if do_observe () then Pp.msgnl strm else () let do_observe_tac s tac g = let goal = Printer.pr_goal g in let lmsg = (str "recdef : ") ++ s in observe (s++fnl()); Stack.push (lmsg,goal) debug_queue; try let v = tac g in ignore(Stack.pop debug_queue); v with e -> if not (Stack.is_empty debug_queue) then begin let e : exn = Cerrors.process_vernac_interp_error e in print_debug_queue true e end ; raise e let observe_tac s tac g = if do_observe () then do_observe_tac s tac g else tac g (* Conclusion tactics *) (* The boolean value is_mes expresses that the termination is expressed using a measure function instead of a well-founded relation. *) let tclUSER tac is_mes l g = let clear_tac = match l with | None -> h_clear false [] | Some l -> tclMAP (fun id -> tclTRY (h_clear false [id])) (List.rev l) in tclTHENSEQ [ clear_tac; if is_mes then tclTHENLIST [ unfold_in_concl [(Termops.all_occurrences, evaluable_of_global_reference (delayed_force Indfun_common.ltof_ref))]; tac ] else tac ] g let tclUSER_if_not_mes concl_tac is_mes names_to_suppress = if is_mes then tclCOMPLETE (h_simplest_apply (delayed_force well_founded_ltof)) else tclUSER concl_tac is_mes names_to_suppress (* Travelling term. Both definitions of [f_terminate] and [f_equation] use the same generic travelling mechanism. *) (* [check_not_nested forbidden e] checks that [e] does not contains any variable of [forbidden] *) let check_not_nested forbidden e = let rec check_not_nested e = match kind_of_term e with | Rel _ -> () | Var x -> if List.mem x (forbidden) then error ("check_not_nested : failure "^string_of_id x) | Meta _ | Evar _ | Sort _ -> () | Cast(e,_,t) -> check_not_nested e;check_not_nested t | Prod(_,t,b) -> check_not_nested t;check_not_nested b | Lambda(_,t,b) -> check_not_nested t;check_not_nested b | LetIn(_,v,t,b) -> check_not_nested t;check_not_nested b;check_not_nested v | App(f,l) -> check_not_nested f;Array.iter check_not_nested l | Const _ -> () | Ind _ -> () | Construct _ -> () | Case(_,t,e,a) -> check_not_nested t;check_not_nested e;Array.iter check_not_nested a | Fix _ -> error "check_not_nested : Fix" | CoFix _ -> error "check_not_nested : Fix" in try check_not_nested e with UserError(_,p) -> errorlabstrm "_" (str "on expr : " ++ Printer.pr_lconstr e ++ str " " ++ p) (* ['a info] contains the local information for travelling *) type 'a infos = { nb_arg : int; (* function number of arguments *) concl_tac : tactic; (* final tactic to finish proofs *) rec_arg_id : identifier; (*name of the declared recursive argument *) is_mes : bool; (* type of recursion *) ih : identifier; (* induction hypothesis name *) f_id : identifier; (* function name *) f_constr : constr; (* function term *) f_terminate : constr; (* termination proof term *) func : global_reference; (* functionnal reference *) info : 'a; is_main_branch : bool; (* on the main branch or on a matched expression *) is_final : bool; (* final first order term or not *) values_and_bounds : (identifier*identifier) list; eqs : identifier list; forbidden_ids : identifier list; acc_inv : constr lazy_t; acc_id : identifier; args_assoc : ((constr list)*constr) list; } type ('a,'b) journey_info_tac = 'a -> (* the arguments of the constructor *) 'b infos -> (* infos of the caller *) ('b infos -> tactic) -> (* the continuation tactic of the caller *) 'b infos -> (* argument of the tactic *) tactic (* journey_info : specifies the actions to do on the different term constructors during the travelling of the term *) type journey_info = { letiN : ((name*constr*types*constr),constr) journey_info_tac; lambdA : ((name*types*constr),constr) journey_info_tac; casE : ((constr infos -> tactic) -> constr infos -> tactic) -> ((case_info * constr * constr * constr array),constr) journey_info_tac; otherS : (unit,constr) journey_info_tac; apP : (constr*(constr list),constr) journey_info_tac; app_reC : (constr*(constr list),constr) journey_info_tac; message : string } let rec add_vars forbidden e = match kind_of_term e with | Var x -> x::forbidden | _ -> fold_constr add_vars forbidden e let treat_case forbid_new_ids to_intros finalize_tac nb_lam e infos : tactic = fun g -> let rev_context,b = decompose_lam_n nb_lam e in let ids = List.fold_left (fun acc (na,_) -> let pre_id = match na with | Name x -> x | Anonymous -> ano_id in pre_id::acc ) [] rev_context in let rev_ids = pf_get_new_ids (List.rev ids) g in let new_b = substl (List.map mkVar rev_ids) b in tclTHENSEQ [ h_intros (List.rev rev_ids); intro_using teq_id; onLastHypId (fun heq -> tclTHENSEQ[ thin to_intros; h_intros to_intros; (fun g' -> let ty_teq = pf_type_of g' (mkVar heq) in let teq_lhs,teq_rhs = let _,args = try destApp ty_teq with _ -> Pp.msgnl (Printer.pr_goal g' ++ fnl () ++ pr_id heq ++ str ":" ++ Printer.pr_lconstr ty_teq); assert false in args.(1),args.(2) in let new_b' = Termops.replace_term teq_lhs teq_rhs new_b in let new_infos = { infos with info = new_b'; eqs = heq::infos.eqs; forbidden_ids = if forbid_new_ids then add_vars infos.forbidden_ids new_b' else infos.forbidden_ids } in finalize_tac new_infos g' ) ] ) ] g let rec travel_aux jinfo continuation_tac (expr_info:constr infos) = match kind_of_term expr_info.info with | CoFix _ | Fix _ -> error "Function cannot treat local fixpoint or cofixpoint" | LetIn(na,b,t,e) -> begin let new_continuation_tac = jinfo.letiN (na,b,t,e) expr_info continuation_tac in travel jinfo new_continuation_tac {expr_info with info = b; is_final=false} end | Rel _ -> anomaly "Free var in goal conclusion !" | Prod _ -> begin try check_not_nested (expr_info.f_id::expr_info.forbidden_ids) expr_info.info; jinfo.otherS () expr_info continuation_tac expr_info with _ -> errorlabstrm "Recdef.travel" (str "the term " ++ Printer.pr_lconstr expr_info.info ++ str " can not contain a recursive call to " ++ pr_id expr_info.f_id) end | Lambda(n,t,b) -> begin try check_not_nested (expr_info.f_id::expr_info.forbidden_ids) expr_info.info; jinfo.otherS () expr_info continuation_tac expr_info with _ -> errorlabstrm "Recdef.travel" (str "the term " ++ Printer.pr_lconstr expr_info.info ++ str " can not contain a recursive call to " ++ pr_id expr_info.f_id) end | Case(ci,t,a,l) -> begin let continuation_tac_a = jinfo.casE (travel jinfo) (ci,t,a,l) expr_info continuation_tac in travel jinfo continuation_tac_a {expr_info with info = a; is_main_branch = false; is_final = false} end | App _ -> let f,args = decompose_app expr_info.info in if eq_constr f (expr_info.f_constr) then jinfo.app_reC (f,args) expr_info continuation_tac expr_info else begin match kind_of_term f with | App _ -> assert false (* f is coming from a decompose_app *) | Const _ | Construct _ | Rel _ | Evar _ | Meta _ | Ind _ | Sort _ | Prod _ | Var _ -> let new_infos = {expr_info with info=(f,args)} in let new_continuation_tac = jinfo.apP (f,args) expr_info continuation_tac in travel_args jinfo expr_info.is_main_branch new_continuation_tac new_infos | _ -> assert false end | Cast(t,_,_) -> travel jinfo continuation_tac {expr_info with info=t} | Const _ | Var _ | Meta _ | Evar _ | Sort _ | Construct _ | Ind _ -> let new_continuation_tac = jinfo.otherS () expr_info continuation_tac in new_continuation_tac expr_info and travel_args jinfo is_final continuation_tac infos = let (f_args',args) = infos.info in match args with | [] -> continuation_tac {infos with info = f_args'; is_final = is_final} | arg::args' -> let new_continuation_tac new_infos = let new_arg = new_infos.info in travel_args jinfo is_final continuation_tac {new_infos with info = (mkApp(f_args',[|new_arg|]),args')} in travel jinfo new_continuation_tac {infos with info=arg;is_final=false} and travel jinfo continuation_tac expr_info = observe_tac (str jinfo.message ++ Printer.pr_lconstr expr_info.info) (travel_aux jinfo continuation_tac expr_info) (* Termination proof *) let rec prove_lt hyple g = begin try let (_,args) = decompose_app (pf_concl g) in let x = try destVar (List.hd args) with _ -> assert false in let z = try destVar (List.hd (List.tl args)) with _ -> assert false in let h = List.find (fun id -> let _,args' = decompose_app (pf_type_of g (mkVar id)) in try x = destVar (List.hd args') with _ -> false ) hyple in let y = List.hd (List.tl (snd (decompose_app (pf_type_of g (mkVar h))))) in tclTHENLIST[ apply (mkApp(le_lt_trans (),[|mkVar x;y;mkVar z;mkVar h|])); observe_tac (str "prove_lt") (prove_lt hyple) ] with Not_found -> ( ( tclTHENLIST[ apply (delayed_force lt_S_n); (observe_tac (str "assumption: " ++ Printer.pr_goal g) (h_assumption)) ]) ) end g let rec destruct_bounds_aux infos (bound,hyple,rechyps) lbounds g = match lbounds with | [] -> let ids = pf_ids_of_hyps g in let s_max = mkApp(delayed_force coq_S, [|bound|]) in let k = next_ident_away_in_goal k_id ids in let ids = k::ids in let h' = next_ident_away_in_goal (h'_id) ids in let ids = h'::ids in let def = next_ident_away_in_goal def_id ids in tclTHENLIST[ split (ImplicitBindings [s_max]); intro_then (fun id -> observe_tac (str "destruct_bounds_aux") (tclTHENS (simplest_case (mkVar id)) [ tclTHENLIST[intro_using h_id; simplest_elim(mkApp(delayed_force lt_n_O,[|s_max|])); default_full_auto]; tclTHENLIST[ observe_tac (str "clearing k ") (clear [id]); h_intros [k;h';def]; observe_tac (str "simple_iter") (simpl_iter onConcl); observe_tac (str "unfold functional") (unfold_in_concl[((true,[1]), evaluable_of_global_reference infos.func)]); observe_tac (str "test" ) ( tclTHENLIST[ list_rewrite true (List.fold_right (fun e acc -> (mkVar e,true)::acc) infos.eqs (List.map (fun e -> (e,true)) rechyps) ); (* list_rewrite true *) (* (List.map (fun e -> (mkVar e,true)) infos.eqs) *) (* ; *) (observe_tac (str "finishing") (tclORELSE h_reflexivity (observe_tac (str "calling prove_lt") (prove_lt hyple))))]) ] ] )) ] g | (_,v_bound)::l -> tclTHENLIST[ simplest_elim (mkVar v_bound); h_clear false [v_bound]; tclDO 2 intro; onNthHypId 1 (fun p_hyp -> (onNthHypId 2 (fun p -> tclTHENLIST[ simplest_elim (mkApp(delayed_force max_constr, [| bound; mkVar p|])); tclDO 3 intro; onNLastHypsId 3 (fun lids -> match lids with [hle2;hle1;pmax] -> destruct_bounds_aux infos ((mkVar pmax), hle1::hle2::hyple,(mkVar p_hyp)::rechyps) l | _ -> assert false) ; ] ) ) ) ] g let destruct_bounds infos = destruct_bounds_aux infos (delayed_force coq_O,[],[]) infos.values_and_bounds let terminate_app f_and_args expr_info continuation_tac infos = if expr_info.is_final && expr_info.is_main_branch then tclTHENLIST[ continuation_tac infos; observe_tac (str "first split") (split (ImplicitBindings [infos.info])); observe_tac (str "destruct_bounds (1)") (destruct_bounds infos) ] else continuation_tac infos let terminate_others _ expr_info continuation_tac infos = if expr_info.is_final && expr_info.is_main_branch then tclTHENLIST[ continuation_tac infos; observe_tac (str "first split") (split (ImplicitBindings [infos.info])); observe_tac (str "destruct_bounds") (destruct_bounds infos) ] else continuation_tac infos let terminate_letin (na,b,t,e) expr_info continuation_tac info = let new_e = subst1 info.info e in let new_forbidden = let forbid = try check_not_nested (expr_info.f_id::expr_info.forbidden_ids) b; true with _ -> false in if forbid then match na with | Anonymous -> info.forbidden_ids | Name id -> id::info.forbidden_ids else info.forbidden_ids in continuation_tac {info with info = new_e; forbidden_ids = new_forbidden} (* This is like the previous one except that it also rewrite on all hypotheses except the ones given in the first argument. All the modified hypotheses are generalized in the process and should be introduced back later; the result is the pair of the tactic and the list of hypotheses that have been generalized and cleared. *) let mkDestructEq : identifier list -> constr -> goal sigma -> tactic * identifier list = fun not_on_hyp expr g -> let hyps = pf_hyps g in let to_revert = Util.map_succeed (fun (id,_,t) -> if List.mem id not_on_hyp || not (Termops.occur_term expr t) then failwith "is_expr_context"; id) hyps in let to_revert_constr = List.rev_map mkVar to_revert in let type_of_expr = pf_type_of g expr in let new_hyps = mkApp(Lazy.force refl_equal, [|type_of_expr; expr|]):: to_revert_constr in tclTHENLIST [h_generalize new_hyps; (fun g2 -> change_in_concl None (pattern_occs [((false,[1]), expr)] (pf_env g2) Evd.empty (pf_concl g2)) g2); simplest_case expr], to_revert let terminate_case next_step (ci,a,t,l) expr_info continuation_tac infos g = let b = try check_not_nested (expr_info.f_id::expr_info.forbidden_ids) a; false with _ -> true in let a' = infos.info in let new_info = {infos with info = mkCase(ci,t,a',l); is_main_branch = expr_info.is_main_branch; is_final = expr_info.is_final} in let destruct_tac,rev_to_thin_intro = mkDestructEq [expr_info.rec_arg_id] a' g in let to_thin_intro = List.rev rev_to_thin_intro in observe_tac (str "treating case " ++ int (Array.length l) ++ spc () ++ Printer.pr_lconstr a') (try (tclTHENS destruct_tac (list_map_i (fun i e -> observe_tac (str "do treat case") (treat_case b to_thin_intro (next_step continuation_tac) ci.ci_cstr_ndecls.(i) e new_info)) 0 (Array.to_list l) )) with | UserError("Refiner.thensn_tac3",_) | UserError("Refiner.tclFAIL_s",_) -> (observe_tac (str "is computable " ++ Printer.pr_lconstr new_info.info) (next_step continuation_tac {new_info with info = nf_betaiotazeta new_info.info} ) )) g let terminate_app_rec (f,args) expr_info continuation_tac _ = List.iter (check_not_nested (expr_info.f_id::expr_info.forbidden_ids)) args; begin try let v = List.assoc args expr_info.args_assoc in let new_infos = {expr_info with info = v} in tclTHENLIST[ continuation_tac new_infos; if expr_info.is_final && expr_info.is_main_branch then tclTHENLIST[ observe_tac (str "first split") (split (ImplicitBindings [new_infos.info])); observe_tac (str "destruct_bounds (3)") (destruct_bounds new_infos) ] else tclIDTAC ] with Not_found -> observe_tac (str "terminate_app_rec not found") (tclTHENS (simplest_elim (mkApp(mkVar expr_info.ih,Array.of_list args))) [ tclTHENLIST[ intro_using rec_res_id; intro; onNthHypId 1 (fun v_bound -> (onNthHypId 2 (fun v -> let new_infos = { expr_info with info = (mkVar v); values_and_bounds = (v,v_bound)::expr_info.values_and_bounds; args_assoc=(args,mkVar v)::expr_info.args_assoc } in tclTHENLIST[ continuation_tac new_infos; if expr_info.is_final && expr_info.is_main_branch then tclTHENLIST[ observe_tac (str "first split") (split (ImplicitBindings [new_infos.info])); observe_tac (str "destruct_bounds (2)") (destruct_bounds new_infos) ] else tclIDTAC ] ) ) ) ]; observe_tac (str "proving decreasing") ( tclTHENS (* proof of args < formal args *) (apply (Lazy.force expr_info.acc_inv)) [ observe_tac (str "h_assumption") h_assumption; tclTHENLIST [ tclTRY(list_rewrite true (List.map (fun e -> mkVar e,true) expr_info.eqs ) ); tclUSER expr_info.concl_tac true (Some ( expr_info.ih::expr_info.acc_id:: (fun (x,y) -> y) (List.split expr_info.values_and_bounds) ) ); ] ]) ]) end let terminate_info = { message = "prove_terminate with term "; letiN = terminate_letin; lambdA = (fun _ _ _ _ -> assert false); casE = terminate_case; otherS = terminate_others; apP = terminate_app; app_reC = terminate_app_rec; } let prove_terminate = travel terminate_info (* Equation proof *) let equation_case next_step (ci,a,t,l) expr_info continuation_tac infos = terminate_case next_step (ci,a,t,l) expr_info continuation_tac infos let rec prove_le g = let x,z = let _,args = decompose_app (pf_concl g) in (List.hd args,List.hd (List.tl args)) in tclFIRST[ h_assumption; apply (delayed_force le_n); begin try let matching_fun = pf_is_matching g (Pattern.PApp(Pattern.PRef (reference_of_constr (le ())),[|Pattern.PVar (destVar x);Pattern.PMeta None|])) in let (h,t) = List.find (fun (_,t) -> matching_fun t) (pf_hyps_types g) in let y = let _,args = decompose_app t in List.hd (List.tl args) in tclTHENLIST[ apply(mkApp(le_trans (),[|x;y;z;mkVar h|])); observe_tac (str "prove_le (rec)") (prove_le) ] with Not_found -> tclFAIL 0 (mt()) end; ] g let rec make_rewrite_list expr_info max = function | [] -> tclIDTAC | (_,p,hp)::l -> observe_tac (str "make_rewrite_list") (tclTHENS (observe_tac (str "rewrite heq on " ++ pr_id p ) ( (fun g -> let t_eq = compute_renamed_type g (mkVar hp) in let k,def = let k_na,_,t = destProd t_eq in let _,_,t = destProd t in let def_na,_,_ = destProd t in Nameops.out_name k_na,Nameops.out_name def_na in general_rewrite_bindings false Termops.all_occurrences true (* dep proofs also: *) true (mkVar hp, ExplicitBindings[dummy_loc,NamedHyp def, expr_info.f_constr;dummy_loc,NamedHyp k, (f_S max)]) false g) ) ) [make_rewrite_list expr_info max l; tclTHENLIST[ (* x < S max proof *) apply (delayed_force le_lt_n_Sm); observe_tac (str "prove_le(2)") prove_le ] ] ) let make_rewrite expr_info l hp max = tclTHENFIRST (observe_tac (str "make_rewrite") (make_rewrite_list expr_info max l)) (observe_tac (str "make_rewrite") (tclTHENS (fun g -> let t_eq = compute_renamed_type g (mkVar hp) in let k,def = let k_na,_,t = destProd t_eq in let _,_,t = destProd t in let def_na,_,_ = destProd t in Nameops.out_name k_na,Nameops.out_name def_na in observe_tac (str "general_rewrite_bindings") (general_rewrite_bindings false Termops.all_occurrences true (* dep proofs also: *) true (mkVar hp, ExplicitBindings[dummy_loc,NamedHyp def, expr_info.f_constr;dummy_loc,NamedHyp k, (f_S (f_S max))]) false) g) [observe_tac(str "make_rewrite finalize") ( (* tclORELSE( h_reflexivity) *) (tclTHENLIST[ simpl_iter onConcl; observe_tac (str "unfold functional") (unfold_in_concl[((true,[1]), evaluable_of_global_reference expr_info.func)]); (list_rewrite true (List.map (fun e -> mkVar e,true) expr_info.eqs)); (observe_tac (str "h_reflexivity") h_reflexivity)])) ; tclTHENLIST[ (* x < S (S max) proof *) apply (delayed_force le_lt_SS); observe_tac (str "prove_le (3)") prove_le ] ]) ) let rec compute_max rew_tac max l = match l with | [] -> rew_tac max | (_,p,_)::l -> tclTHENLIST[ simplest_elim (mkApp(delayed_force max_constr, [| max; mkVar p|])); tclDO 3 intro; onNLastHypsId 3 (fun lids -> match lids with | [hle2;hle1;pmax] -> compute_max rew_tac (mkVar pmax) l | _ -> assert false )] let rec destruct_hex expr_info acc l = match l with | [] -> begin match List.rev acc with | [] -> tclIDTAC | (_,p,hp)::tl -> observe_tac (str "compute max ") (compute_max (make_rewrite expr_info tl hp) (mkVar p) tl) end | (v,hex)::l -> tclTHENLIST[ simplest_case (mkVar hex); clear [hex]; tclDO 2 intro; onNthHypId 1 (fun hp -> onNthHypId 2 (fun p -> observe_tac (str "destruct_hex after " ++ pr_id hp ++ spc () ++ pr_id p) (destruct_hex expr_info ((v,p,hp)::acc) l) ) ) ] let rec intros_values_eq expr_info acc = tclORELSE( tclTHENLIST[ tclDO 2 intro; onNthHypId 1 (fun hex -> (onNthHypId 2 (fun v -> intros_values_eq expr_info ((v,hex)::acc))) ) ]) (tclCOMPLETE ( destruct_hex expr_info [] acc )) let equation_others _ expr_info continuation_tac infos = if expr_info.is_final && expr_info.is_main_branch then tclTHEN (continuation_tac infos) (intros_values_eq expr_info []) else continuation_tac infos let equation_letin (na,b,t,e) expr_info continuation_tac info = let new_e = subst1 info.info e in continuation_tac {info with info = new_e;} let equation_app f_and_args expr_info continuation_tac infos = if expr_info.is_final && expr_info.is_main_branch then intros_values_eq expr_info [] else continuation_tac infos let equation_app_rec (f,args) expr_info continuation_tac info = begin try let v = List.assoc args expr_info.args_assoc in let new_infos = {expr_info with info = v} in observe_tac (str "app_rec found") (continuation_tac new_infos) with Not_found -> if expr_info.is_final && expr_info.is_main_branch then tclTHENLIST [ simplest_case (mkApp (expr_info.f_terminate,Array.of_list args)); continuation_tac {expr_info with args_assoc = (args,delayed_force coq_O)::expr_info.args_assoc}; observe_tac (str "app_rec intros_values_eq") (intros_values_eq expr_info []) ] else tclTHENLIST[ simplest_case (mkApp (expr_info.f_terminate,Array.of_list args)); observe_tac (str "app_rec not_found") (continuation_tac {expr_info with args_assoc = (args,delayed_force coq_O)::expr_info.args_assoc}) ] end let equation_info = {message = "prove_equation with term "; letiN = (fun _ -> assert false); lambdA = (fun _ _ _ _ -> assert false); casE = equation_case; otherS = equation_others; apP = equation_app; app_reC = equation_app_rec } let prove_eq = travel equation_info (* wrappers *) (* [compute_terminate_type] computes the type of the Definition f_terminate from the type of f_F *) let compute_terminate_type nb_args func = let _,a_arrow_b,_ = destLambda(def_of_const (constr_of_global func)) in let rev_args,b = decompose_prod_n nb_args a_arrow_b in let left = mkApp(delayed_force iter, Array.of_list (lift 5 a_arrow_b:: mkRel 3:: constr_of_global func::mkRel 1:: List.rev (list_map_i (fun i _ -> mkRel (6+i)) 0 rev_args) ) ) in let right = mkRel 5 in let equality = mkApp(delayed_force eq, [|lift 5 b; left; right|]) in let result = (mkProd ((Name def_id) , lift 4 a_arrow_b, equality)) in let cond = mkApp(delayed_force lt, [|(mkRel 2); (mkRel 1)|]) in let nb_iter = mkApp(delayed_force ex, [|delayed_force nat; (mkLambda (Name p_id, delayed_force nat, (mkProd (Name k_id, delayed_force nat, mkArrow cond result))))|])in let value = mkApp(delayed_force coq_sig, [|b; (mkLambda (Name v_id, b, nb_iter))|]) in compose_prod rev_args value let termination_proof_header is_mes input_type ids args_id relation rec_arg_num rec_arg_id tac wf_tac : tactic = begin fun g -> let nargs = List.length args_id in let pre_rec_args = List.rev_map mkVar (fst (list_chop (rec_arg_num - 1) args_id)) in let relation = substl pre_rec_args relation in let input_type = substl pre_rec_args input_type in let wf_thm = next_ident_away_in_goal (id_of_string ("wf_R")) ids in let wf_rec_arg = next_ident_away_in_goal (id_of_string ("Acc_"^(string_of_id rec_arg_id))) (wf_thm::ids) in let hrec = next_ident_away_in_goal hrec_id (wf_rec_arg::wf_thm::ids) in let acc_inv = lazy ( mkApp ( delayed_force acc_inv_id, [|input_type;relation;mkVar rec_arg_id|] ) ) in tclTHEN (h_intros args_id) (tclTHENS (observe_tac (str "first assert") (assert_tac (Name wf_rec_arg) (mkApp (delayed_force acc_rel, [|input_type;relation;mkVar rec_arg_id|]) ) ) ) [ (* accesibility proof *) tclTHENS (observe_tac (str "second assert") (assert_tac (Name wf_thm) (mkApp (delayed_force well_founded,[|input_type;relation|])) ) ) [ (* interactive proof that the relation is well_founded *) observe_tac (str "wf_tac") (wf_tac is_mes (Some args_id)); (* this gives the accessibility argument *) observe_tac (str "apply wf_thm") (h_simplest_apply (mkApp(mkVar wf_thm,[|mkVar rec_arg_id|])) ) ] ; (* rest of the proof *) tclTHENSEQ [observe_tac (str "generalize") (onNLastHypsId (nargs+1) (tclMAP (fun id -> tclTHEN (h_generalize [mkVar id]) (h_clear false [id])) )) ; observe_tac (str "h_fix") (h_fix (Some hrec) (nargs+1)); h_intros args_id; h_intro wf_rec_arg; observe_tac (str "tac") (tac wf_rec_arg hrec wf_rec_arg acc_inv) ] ] ) g end let rec instantiate_lambda t l = match l with | [] -> t | a::l -> let (_, _, body) = destLambda t in instantiate_lambda (subst1 a body) l let whole_start (concl_tac:tactic) nb_args is_mes func input_type relation rec_arg_num : tactic = begin fun g -> let ids = Termops.ids_of_named_context (pf_hyps g) in let func_body = (def_of_const (constr_of_global func)) in let (f_name, _, body1) = destLambda func_body in let f_id = match f_name with | Name f_id -> next_ident_away_in_goal f_id ids | Anonymous -> anomaly "Anonymous function" in let n_names_types,_ = decompose_lam_n nb_args body1 in let n_ids,ids = List.fold_left (fun (n_ids,ids) (n_name,_) -> match n_name with | Name id -> let n_id = next_ident_away_in_goal id ids in n_id::n_ids,n_id::ids | _ -> anomaly "anonymous argument" ) ([],(f_id::ids)) n_names_types in let rec_arg_id = List.nth n_ids (rec_arg_num - 1) in let expr = instantiate_lambda func_body (mkVar f_id::(List.map mkVar n_ids)) in termination_proof_header is_mes input_type ids n_ids relation rec_arg_num rec_arg_id (fun rec_arg_id hrec acc_id acc_inv g -> (prove_terminate (fun infos -> tclIDTAC) { is_main_branch = true; (* we are on the main branche (i.e. still on a match ... with .... end *) is_final = true; (* and on leaf (more or less) *) f_terminate = delayed_force coq_O; nb_arg = nb_args; concl_tac = concl_tac; rec_arg_id = rec_arg_id; is_mes = is_mes; ih = hrec; f_id = f_id; f_constr = mkVar f_id; func = func; info = expr; acc_inv = acc_inv; acc_id = acc_id; values_and_bounds = []; eqs = []; forbidden_ids = []; args_assoc = [] } ) g ) (tclUSER_if_not_mes concl_tac) g end let get_current_subgoals_types () = let p = Proof_global.give_me_the_proof () in let { Evd.it=sgs ; sigma=sigma } = Proof.V82.subgoals p in List.map (Goal.V82.abstract_type sigma) sgs let build_and_l l = let and_constr = Coqlib.build_coq_and () in let conj_constr = coq_conj () in let mk_and p1 p2 = Term.mkApp(and_constr,[|p1;p2|]) in let rec is_well_founded t = match kind_of_term t with | Prod(_,_,t') -> is_well_founded t' | App(_,_) -> let (f,_) = decompose_app t in eq_constr f (well_founded ()) | _ -> assert false in let compare t1 t2 = let b1,b2= is_well_founded t1,is_well_founded t2 in if (b1&&b2) || not (b1 || b2) then 0 else if b1 && not b2 then 1 else -1 in let l = List.sort compare l in let rec f = function | [] -> failwith "empty list of subgoals!" | [p] -> p,tclIDTAC,1 | p1::pl -> let c,tac,nb = f pl in mk_and p1 c, tclTHENS (apply (constr_of_global conj_constr)) [tclIDTAC; tac ],nb+1 in f l let is_rec_res id = let rec_res_name = string_of_id rec_res_id in let id_name = string_of_id id in try String.sub id_name 0 (String.length rec_res_name) = rec_res_name with _ -> false let clear_goals = let rec clear_goal t = match kind_of_term t with | Prod(Name id as na,t',b) -> let b' = clear_goal b in if noccurn 1 b' && (is_rec_res id) then Termops.pop b' else if b' == b then t else mkProd(na,t',b') | _ -> map_constr clear_goal t in List.map clear_goal let build_new_goal_type () = let sub_gls_types = get_current_subgoals_types () in (* Pp.msgnl (str "sub_gls_types1 := " ++ Util.prlist_with_sep (fun () -> Pp.fnl () ++ Pp.fnl ()) Printer.pr_lconstr sub_gls_types); *) let sub_gls_types = clear_goals sub_gls_types in (* Pp.msgnl (str "sub_gls_types2 := " ++ Util.prlist_with_sep (fun () -> Pp.fnl () ++ Pp.fnl ()) Printer.pr_lconstr sub_gls_types); *) let res = build_and_l sub_gls_types in res let is_opaque_constant c = let cb = Global.lookup_constant c in match cb.Declarations.const_body with | Declarations.OpaqueDef _ -> true | Declarations.Undef _ -> true | Declarations.Def _ -> false let open_new_goal (build_proof:tactic -> tactic -> unit) using_lemmas ref_ goal_name (gls_type,decompose_and_tac,nb_goal) = (* Pp.msgnl (str "gls_type := " ++ Printer.pr_lconstr gls_type); *) let current_proof_name = get_current_proof_name () in let name = match goal_name with | Some s -> s | None -> try (add_suffix current_proof_name "_subproof") with _ -> anomaly "open_new_goal with an unamed theorem" in let sign = initialize_named_context_for_proof () in let na = next_global_ident_away name [] in if Termops.occur_existential gls_type then Errors.error "\"abstract\" cannot handle existentials"; let hook _ _ = let opacity = let na_ref = Libnames.Ident (dummy_loc,na) in let na_global = Nametab.global na_ref in match na_global with ConstRef c -> is_opaque_constant c | _ -> anomaly "equation_lemma: not a constant" in let lemma = mkConst (Lib.make_con na) in ref_ := Some lemma ; let lid = ref [] in let h_num = ref (-1) in Flags.silently Vernacentries.interp (Vernacexpr.VernacAbort None); build_proof ( fun gls -> let hid = next_ident_away_in_goal h_id (pf_ids_of_hyps gls) in tclTHENSEQ [ h_generalize [lemma]; h_intro hid; (fun g -> let ids = pf_ids_of_hyps g in tclTHEN (Elim.h_decompose_and (mkVar hid)) (fun g -> let ids' = pf_ids_of_hyps g in lid := List.rev (list_subtract ids' ids); if !lid = [] then lid := [hid]; tclIDTAC g ) g ); ] gls) (fun g -> match kind_of_term (pf_concl g) with | App(f,_) when eq_constr f (well_founded ()) -> Auto.h_auto None [] (Some []) g | _ -> incr h_num; (observe_tac (str "finishing using") ( tclCOMPLETE( tclFIRST[ tclTHEN (eapply_with_bindings (mkVar (List.nth !lid !h_num), NoBindings)) e_assumption; Eauto.eauto_with_bases false (true,5) [Evd.empty,Lazy.force refl_equal] [Auto.Hint_db.empty empty_transparent_state false] ] ) ) ) g) ; Lemmas.save_named opacity; in start_proof na (Decl_kinds.Global, Decl_kinds.Proof Decl_kinds.Lemma) sign gls_type hook ; if Indfun_common.is_strict_tcc () then by (tclIDTAC) else begin by ( fun g -> tclTHEN (decompose_and_tac) (tclORELSE (tclFIRST (List.map (fun c -> tclTHENSEQ [intros; h_simplest_apply (interp_constr Evd.empty (Global.env()) c); tclCOMPLETE Auto.default_auto ] ) using_lemmas) ) tclIDTAC) g) end; try by tclIDTAC; (* raises UserError _ if the proof is complete *) if Flags.is_verbose () then (pp (Printer.pr_open_subgoals())) with UserError _ -> defined () let com_terminate tcc_lemma_name tcc_lemma_ref is_mes fonctional_ref input_type relation rec_arg_num thm_name using_lemmas nb_args hook = let start_proof (tac_start:tactic) (tac_end:tactic) = let (evmap, env) = Lemmas.get_current_context() in start_proof thm_name (Global, Proof Lemma) (Environ.named_context_val env) (compute_terminate_type nb_args fonctional_ref) hook; by (observe_tac (str "starting_tac") tac_start); by (observe_tac (str "whole_start") (whole_start tac_end nb_args is_mes fonctional_ref input_type relation rec_arg_num )) in start_proof tclIDTAC tclIDTAC; try let new_goal_type = build_new_goal_type () in open_new_goal start_proof using_lemmas tcc_lemma_ref (Some tcc_lemma_name) (new_goal_type); with Failure "empty list of subgoals!" -> (* a non recursive function declared with measure ! *) defined () let start_equation (f:global_reference) (term_f:global_reference) (cont_tactic:identifier list -> tactic) g = let ids = pf_ids_of_hyps g in let terminate_constr = constr_of_global term_f in let nargs = nb_prod (type_of_const terminate_constr) in let x = n_x_id ids nargs in tclTHENLIST [ h_intros x; unfold_in_concl [(Termops.all_occurrences, evaluable_of_global_reference f)]; observe_tac (str "simplest_case") (simplest_case (mkApp (terminate_constr, Array.of_list (List.map mkVar x)))); observe_tac (str "prove_eq") (cont_tactic x)] g;; let (com_eqn : int -> identifier -> global_reference -> global_reference -> global_reference -> constr -> unit) = fun nb_arg eq_name functional_ref f_ref terminate_ref equation_lemma_type -> let opacity = match terminate_ref with | ConstRef c -> is_opaque_constant c | _ -> anomaly "terminate_lemma: not a constant" in let (evmap, env) = Lemmas.get_current_context() in let f_constr = constr_of_global f_ref in let equation_lemma_type = subst1 f_constr equation_lemma_type in (start_proof eq_name (Global, Proof Lemma) (Environ.named_context_val env) equation_lemma_type (fun _ _ -> ()); by (start_equation f_ref terminate_ref (fun x -> prove_eq (fun _ -> tclIDTAC) {nb_arg=nb_arg; f_terminate = constr_of_global terminate_ref; f_constr = f_constr; concl_tac = tclIDTAC; func=functional_ref; info=(instantiate_lambda (def_of_const (constr_of_global functional_ref)) (f_constr::List.map mkVar x) ); is_main_branch = true; is_final = true; values_and_bounds = []; eqs = []; forbidden_ids = []; acc_inv = lazy (assert false); acc_id = id_of_string "____"; args_assoc = []; f_id = id_of_string "______"; rec_arg_id = id_of_string "______"; is_mes = false; ih = id_of_string "______"; } ) ); (* (try Vernacentries.interp (Vernacexpr.VernacShow Vernacexpr.ShowProof) with _ -> ()); *) (* Vernacentries.interp (Vernacexpr.VernacShow Vernacexpr.ShowScript); *) Flags.silently (fun () -> Lemmas.save_named opacity) () ; (* Pp.msgnl (str "eqn finished"); *) );; let recursive_definition is_mes function_name rec_impls type_of_f r rec_arg_num eq generate_induction_principle using_lemmas : unit = let previous_label = Lib.current_command_label () in let function_type = interp_constr Evd.empty (Global.env()) type_of_f in let env = push_named (function_name,None,function_type) (Global.env()) in (* Pp.msgnl (str "function type := " ++ Printer.pr_lconstr function_type); *) let equation_lemma_type = nf_betaiotazeta (interp_gen (OfType None) Evd.empty env ~impls:rec_impls eq) in (* Pp.msgnl (str "lemma type := " ++ Printer.pr_lconstr equation_lemma_type ++ fnl ()); *) let res_vars,eq' = decompose_prod equation_lemma_type in let env_eq' = Environ.push_rel_context (List.map (fun (x,y) -> (x,None,y)) res_vars) env in let eq' = nf_zeta env_eq' eq' in let res = (* Pp.msgnl (str "res_var :=" ++ Printer.pr_lconstr_env (push_rel_context (List.map (function (x,t) -> (x,None,t)) res_vars) env) eq'); *) (* Pp.msgnl (str "rec_arg_num := " ++ str (string_of_int rec_arg_num)); *) (* Pp.msgnl (str "eq' := " ++ str (string_of_int rec_arg_num)); *) match kind_of_term eq' with | App(e,[|_;_;eq_fix|]) -> mkLambda (Name function_name,function_type,subst_var function_name (compose_lam res_vars eq_fix)) | _ -> failwith "Recursive Definition (res not eq)" in let pre_rec_args,function_type_before_rec_arg = decompose_prod_n (rec_arg_num - 1) function_type in let (_, rec_arg_type, _) = destProd function_type_before_rec_arg in let arg_types = List.rev_map snd (fst (decompose_prod_n (List.length res_vars) function_type)) in let equation_id = add_suffix function_name "_equation" in let functional_id = add_suffix function_name "_F" in let term_id = add_suffix function_name "_terminate" in let functional_ref = declare_fun functional_id (IsDefinition Decl_kinds.Definition) res in let env_with_pre_rec_args = push_rel_context(List.map (function (x,t) -> (x,None,t)) pre_rec_args) env in let relation = interp_constr Evd.empty env_with_pre_rec_args r in let tcc_lemma_name = add_suffix function_name "_tcc" in let tcc_lemma_constr = ref None in (* let _ = Pp.msgnl (str "relation := " ++ Printer.pr_lconstr_env env_with_pre_rec_args relation) in *) let hook _ _ = let term_ref = Nametab.locate (qualid_of_ident term_id) in let f_ref = declare_f function_name (IsProof Lemma) arg_types term_ref in let _ = Table.extraction_inline true [Ident (dummy_loc,term_id)] in (* message "start second proof"; *) let stop = try com_eqn (List.length res_vars) equation_id functional_ref f_ref term_ref (subst_var function_name equation_lemma_type); false with e -> begin if do_observe () then pperrnl (str "Cannot create equation Lemma " ++ Errors.print e) else anomaly "Cannot create equation Lemma" ; true end in if not stop then let eq_ref = Nametab.locate (qualid_of_ident equation_id ) in let f_ref = destConst (constr_of_global f_ref) and functional_ref = destConst (constr_of_global functional_ref) and eq_ref = destConst (constr_of_global eq_ref) in generate_induction_principle f_ref tcc_lemma_constr functional_ref eq_ref rec_arg_num rec_arg_type (nb_prod res) relation; if Flags.is_verbose () then msgnl (h 1 (Ppconstr.pr_id function_name ++ spc () ++ str"is defined" )++ fnl () ++ h 1 (Ppconstr.pr_id equation_id ++ spc () ++ str"is defined" ) ) in try com_terminate tcc_lemma_name tcc_lemma_constr is_mes functional_ref rec_arg_type relation rec_arg_num term_id using_lemmas (List.length res_vars) hook with e -> begin (try ignore (Backtrack.backto previous_label) with _ -> ()); (* anomaly "Cannot create termination Lemma" *) raise e end