(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* hov 1 (Ppconstr.pr_id id ++ str " := " ++ Pp.cut () ++ prc c) | loc, AnonHyp n, c -> hov 1 (int n ++ str " := " ++ Pp.cut () ++ prc c) let pr_bindings prc prlc = function | ImplicitBindings l -> brk (1,1) ++ str "with" ++ brk (1,1) ++ pr_sequence prc l | ExplicitBindings l -> brk (1,1) ++ str "with" ++ brk (1,1) ++ pr_sequence (fun b -> str"(" ++ pr_binding prlc b ++ str")") l | NoBindings -> mt () let pr_with_bindings prc prlc (c,bl) = prc c ++ hv 0 (pr_bindings prc prlc bl) let pr_constr_with_binding prc (c,bl) : Pp.std_ppcmds = pr_with_bindings prc prc (c,bl) (* The local debugging mechanism *) (* let msgnl = Pp.msgnl *) let observe strm = if do_observe () then Pp.msg_debug strm else () (*let observennl strm = if do_observe () then begin Pp.msg strm;Pp.pp_flush () end else ()*) let do_observe_tac s tac g = let goal = try Printer.pr_goal g with e when Errors.noncritical e -> assert false in try let v = tac g in msgnl (goal ++ fnl () ++ s ++(str " ")++(str "finished")); v with reraise -> let reraise = Errors.push reraise in let e = Cerrors.process_vernac_interp_error reraise in observe (str "observation "++ s++str " raised exception " ++ Errors.iprint e ++ str " on goal " ++ goal ); iraise reraise;; let observe_tac_strm s tac g = if do_observe () then do_observe_tac s tac g else tac g let observe_tac s tac g = if do_observe () then do_observe_tac (str s) tac g else tac g (* [nf_zeta] $\zeta$-normalization of a term *) let nf_zeta = Reductionops.clos_norm_flags (Closure.RedFlags.mkflags [Closure.RedFlags.fZETA]) Environ.empty_env Evd.empty (* (\* [id_to_constr id] finds the term associated to [id] in the global environment *\) *) (* let id_to_constr id = *) (* try *) (* Constrintern.global_reference id *) (* with Not_found -> *) (* raise (UserError ("",str "Cannot find " ++ Ppconstr.pr_id id)) *) let make_eq () = try Universes.constr_of_global (Coqlib.build_coq_eq ()) with _ -> assert false let make_eq_refl () = try Universes.constr_of_global (Coqlib.build_coq_eq_refl ()) with _ -> assert false (* [generate_type g_to_f f graph i] build the completeness (resp. correctness) lemma type if [g_to_f = true] (resp. g_to_f = false) where [graph] is the graph of [f] and is the [i]th function in the block. [generate_type true f i] returns \[\forall (x_1:t_1)\ldots(x_n:t_n), let fv := f x_1\ldots x_n in, forall res, graph\ x_1\ldots x_n\ res \rightarrow res = fv \] decomposed as the context and the conclusion [generate_type false f i] returns \[\forall (x_1:t_1)\ldots(x_n:t_n), let fv := f x_1\ldots x_n in, forall res, res = fv \rightarrow graph\ x_1\ldots x_n\ res\] decomposed as the context and the conclusion *) let generate_type evd g_to_f f graph i = (*i we deduce the number of arguments of the function and its returned type from the graph i*) let evd',graph = Evd.fresh_global (Global.env ()) !evd (Globnames.IndRef (fst (destInd graph))) in evd:=evd'; let graph_arity = Typing.e_type_of (Global.env ()) evd graph in let ctxt,_ = decompose_prod_assum graph_arity in let fun_ctxt,res_type = match ctxt with | [] | [_] -> anomaly (Pp.str "Not a valid context") | (_,_,res_type)::fun_ctxt -> fun_ctxt,res_type in let rec args_from_decl i accu = function | [] -> accu | (_, Some _, _) :: l -> args_from_decl (succ i) accu l | _ :: l -> let t = mkRel i in args_from_decl (succ i) (t :: accu) l in (*i We need to name the vars [res] and [fv] i*) let filter = function (Name id,_,_) -> Some id | (Anonymous,_,_) -> None in let named_ctxt = List.map_filter filter fun_ctxt in let res_id = Namegen.next_ident_away_in_goal (Id.of_string "_res") named_ctxt in let fv_id = Namegen.next_ident_away_in_goal (Id.of_string "fv") (res_id :: named_ctxt) in (*i we can then type the argument to be applied to the function [f] i*) let args_as_rels = Array.of_list (args_from_decl 1 [] fun_ctxt) in (*i the hypothesis [res = fv] can then be computed We will need to lift it by one in order to use it as a conclusion i*) let make_eq = make_eq () in let res_eq_f_of_args = mkApp(make_eq ,[|lift 2 res_type;mkRel 1;mkRel 2|]) in (*i The hypothesis [graph\ x_1\ldots x_n\ res] can then be computed We will need to lift it by one in order to use it as a conclusion i*) let args_and_res_as_rels = Array.of_list (args_from_decl 3 [] fun_ctxt) in let args_and_res_as_rels = Array.append args_and_res_as_rels [|mkRel 1|] in let graph_applied = mkApp(graph, args_and_res_as_rels) in (*i The [pre_context] is the defined to be the context corresponding to \[\forall (x_1:t_1)\ldots(x_n:t_n), let fv := f x_1\ldots x_n in, forall res, \] i*) let pre_ctxt = (Name res_id,None,lift 1 res_type)::(Name fv_id,Some (mkApp(f,args_as_rels)),res_type)::fun_ctxt in (*i and we can return the solution depending on which lemma type we are defining i*) if g_to_f then (Anonymous,None,graph_applied)::pre_ctxt,(lift 1 res_eq_f_of_args),graph else (Anonymous,None,res_eq_f_of_args)::pre_ctxt,(lift 1 graph_applied),graph (* [find_induction_principle f] searches and returns the [body] and the [type] of [f_rect] WARNING: while convertible, [type_of body] and [type] can be non equal *) let find_induction_principle evd f = let f_as_constant,u = match kind_of_term f with | Const c' -> c' | _ -> error "Must be used with a function" in let infos = find_Function_infos f_as_constant in match infos.rect_lemma with | None -> raise Not_found | Some rect_lemma -> let evd',rect_lemma = Evd.fresh_global (Global.env ()) !evd (Globnames.ConstRef rect_lemma) in let evd',typ = Typing.type_of ~refresh:true (Global.env ()) evd' rect_lemma in evd:=evd'; rect_lemma,typ let rec generate_fresh_id x avoid i = if i == 0 then [] else let id = Namegen.next_ident_away_in_goal x avoid in id::(generate_fresh_id x (id::avoid) (pred i)) (* [prove_fun_correct functional_induction funs_constr graphs_constr schemes lemmas_types_infos i ] is the tactic used to prove correctness lemma. [functional_induction] is the tactic defined in [indfun] (dependency problem) [funs_constr], [graphs_constr] [schemes] [lemmas_types_infos] are the mutually recursive functions (resp. graphs of the functions and principles and correctness lemma types) to prove correct. [i] is the indice of the function to prove correct The lemma to prove if suppose to have been generated by [generate_type] (in $\zeta$ normal form that is it looks like~: [\forall (x_1:t_1)\ldots(x_n:t_n), forall res, res = f x_1\ldots x_n in, \rightarrow graph\ x_1\ldots x_n\ res] The sketch of the proof is the following one~: \begin{enumerate} \item intros until $x_n$ \item $functional\ induction\ (f.(i)\ x_1\ldots x_n)$ using schemes.(i) \item for each generated branch intro [res] and [hres :res = f x_1\ldots x_n], rewrite [hres] and the apply the corresponding constructor of the corresponding graph inductive. \end{enumerate} *) let prove_fun_correct evd functional_induction funs_constr graphs_constr schemes lemmas_types_infos i : tactic = fun g -> (* first of all we recreate the lemmas types to be used as predicates of the induction principle that is~: \[fun (x_1:t_1)\ldots(x_n:t_n)=> fun fv => fun res => res = fv \rightarrow graph\ x_1\ldots x_n\ res\] *) (* we the get the definition of the graphs block *) let graph_ind,u = destInd graphs_constr.(i) in let kn = fst graph_ind in let mib,_ = Global.lookup_inductive graph_ind in (* and the principle to use in this lemma in $\zeta$ normal form *) let f_principle,princ_type = schemes.(i) in let princ_type = nf_zeta princ_type in let princ_infos = Tactics.compute_elim_sig princ_type in (* The number of args of the function is then easily computable *) let nb_fun_args = nb_prod (pf_concl g) - 2 in let args_names = generate_fresh_id (Id.of_string "x") [] nb_fun_args in let ids = args_names@(pf_ids_of_hyps g) in (* Since we cannot ensure that the functional principle is defined in the environment and due to the bug #1174, we will need to pose the principle using a name *) let principle_id = Namegen.next_ident_away_in_goal (Id.of_string "princ") ids in let ids = principle_id :: ids in (* We get the branches of the principle *) let branches = List.rev princ_infos.branches in (* and built the intro pattern for each of them *) let intro_pats = List.map (fun (_,_,br_type) -> List.map (fun id -> Loc.ghost, IntroNaming (IntroIdentifier id)) (generate_fresh_id (Id.of_string "y") ids (List.length (fst (decompose_prod_assum br_type)))) ) branches in (* before building the full intro pattern for the principle *) let eq_ind = make_eq () in let eq_construct = mkConstructUi (destInd eq_ind, 1) in (* The next to referencies will be used to find out which constructor to apply in each branch *) let ind_number = ref 0 and min_constr_number = ref 0 in (* The tactic to prove the ith branch of the principle *) let prove_branche i g = (* We get the identifiers of this branch *) let pre_args = List.fold_right (fun (_,pat) acc -> match pat with | IntroNaming (IntroIdentifier id) -> id::acc | _ -> anomaly (Pp.str "Not an identifier") ) (List.nth intro_pats (pred i)) [] in (* and get the real args of the branch by unfolding the defined constant *) (* We can then recompute the arguments of the constructor. For each [hid] introduced by this branch, if [hid] has type $forall res, res=fv -> graph.(j)\ x_1\ x_n res$ the corresponding arguments of the constructor are [ fv (hid fv (refl_equal fv)) ]. If [hid] has another type the corresponding argument of the constructor is [hid] *) let constructor_args g = List.fold_right (fun hid acc -> let type_of_hid = pf_unsafe_type_of g (mkVar hid) in match kind_of_term type_of_hid with | Prod(_,_,t') -> begin match kind_of_term t' with | Prod(_,t'',t''') -> begin match kind_of_term t'',kind_of_term t''' with | App(eq,args), App(graph',_) when (eq_constr eq eq_ind) && Array.exists (Constr.eq_constr_nounivs graph') graphs_constr -> (args.(2)::(mkApp(mkVar hid,[|args.(2);(mkApp(eq_construct,[|args.(0);args.(2)|]))|])) ::acc) | _ -> mkVar hid :: acc end | _ -> mkVar hid :: acc end | _ -> mkVar hid :: acc ) pre_args [] in (* in fact we must also add the parameters to the constructor args *) let constructor_args g = let params_id = fst (List.chop princ_infos.nparams args_names) in (List.map mkVar params_id)@((constructor_args g)) in (* We then get the constructor corresponding to this branch and modifies the references has needed i.e. if the constructor is the last one of the current inductive then add one the number of the inductive to take and add the number of constructor of the previous graph to the minimal constructor number *) let constructor = let constructor_num = i - !min_constr_number in let length = Array.length (mib.Declarations.mind_packets.(!ind_number).Declarations.mind_consnames) in if constructor_num <= length then begin (kn,!ind_number),constructor_num end else begin incr ind_number; min_constr_number := !min_constr_number + length ; (kn,!ind_number),1 end in (* we can then build the final proof term *) let app_constructor g = applist((mkConstructU(constructor,u)),constructor_args g) in (* an apply the tactic *) let res,hres = match generate_fresh_id (Id.of_string "z") (ids(* @this_branche_ids *)) 2 with | [res;hres] -> res,hres | _ -> assert false in (* observe (str "constructor := " ++ Printer.pr_lconstr_env (pf_env g) app_constructor); *) ( tclTHENSEQ [ observe_tac("h_intro_patterns ") (let l = (List.nth intro_pats (pred i)) in match l with | [] -> tclIDTAC | _ -> Proofview.V82.of_tactic (intro_patterns l)); (* unfolding of all the defined variables introduced by this branch *) (* observe_tac "unfolding" pre_tac; *) (* $zeta$ normalizing of the conclusion *) reduce (Genredexpr.Cbv { Redops.all_flags with Genredexpr.rDelta = false ; Genredexpr.rConst = [] } ) Locusops.onConcl; observe_tac ("toto ") tclIDTAC; (* introducing the the result of the graph and the equality hypothesis *) observe_tac "introducing" (tclMAP (fun x -> Proofview.V82.of_tactic (Simple.intro x)) [res;hres]); (* replacing [res] with its value *) observe_tac "rewriting res value" (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar hres))); (* Conclusion *) observe_tac "exact" (fun g -> Proofview.V82.of_tactic (exact_check (app_constructor g)) g) ] ) g in (* end of branche proof *) let lemmas = Array.map (fun ((_,(ctxt,concl))) -> match ctxt with | [] | [_] | [_;_] -> anomaly (Pp.str "bad context") | hres::res::(x,_,t)::ctxt -> let res = Termops.it_mkLambda_or_LetIn (Termops.it_mkProd_or_LetIn concl [hres;res]) ((x,None,t)::ctxt) in res ) lemmas_types_infos in let param_names = fst (List.chop princ_infos.nparams args_names) in let params = List.map mkVar param_names in let lemmas = Array.to_list (Array.map (fun c -> applist(c,params)) lemmas) in (* The bindings of the principle that is the params of the principle and the different lemma types *) let bindings = let params_bindings,avoid = List.fold_left2 (fun (bindings,avoid) (x,_,_) p -> let id = Namegen.next_ident_away (Nameops.out_name x) avoid in p::bindings,id::avoid ) ([],pf_ids_of_hyps g) princ_infos.params (List.rev params) in let lemmas_bindings = List.rev (fst (List.fold_left2 (fun (bindings,avoid) (x,_,_) p -> let id = Namegen.next_ident_away (Nameops.out_name x) avoid in (nf_zeta p)::bindings,id::avoid) ([],avoid) princ_infos.predicates (lemmas))) in (params_bindings@lemmas_bindings) in tclTHENSEQ [ observe_tac "principle" (Proofview.V82.of_tactic (assert_by (Name principle_id) princ_type (exact_check f_principle))); observe_tac "intro args_names" (tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) args_names); (* observe_tac "titi" (pose_proof (Name (Id.of_string "__")) (Reductionops.nf_beta Evd.empty ((mkApp (mkVar principle_id,Array.of_list bindings))))); *) observe_tac "idtac" tclIDTAC; tclTHEN_i (observe_tac "functional_induction" ( (fun gl -> let term = mkApp (mkVar principle_id,Array.of_list bindings) in let gl', _ty = pf_eapply (Typing.type_of ~refresh:true) gl term in Proofview.V82.of_tactic (apply term) gl') )) (fun i g -> observe_tac ("proving branche "^string_of_int i) (prove_branche i) g ) ] g (* [generalize_dependent_of x hyp g] generalize every hypothesis which depends of [x] but [hyp] *) let generalize_dependent_of x hyp g = tclMAP (function | (id,None,t) when not (Id.equal id hyp) && (Termops.occur_var (pf_env g) x t) -> tclTHEN (Tactics.Simple.generalize [mkVar id]) (thin [id]) | _ -> tclIDTAC ) (pf_hyps g) g (* [intros_with_rewrite] do the intros in each branch and treat each new hypothesis (unfolding, substituting, destructing cases \ldots) *) let rec intros_with_rewrite g = observe_tac "intros_with_rewrite" intros_with_rewrite_aux g and intros_with_rewrite_aux : tactic = fun g -> let eq_ind = make_eq () in match kind_of_term (pf_concl g) with | Prod(_,t,t') -> begin match kind_of_term t with | App(eq,args) when (eq_constr eq eq_ind) -> if Reductionops.is_conv (pf_env g) (project g) args.(1) args.(2) then let id = pf_get_new_id (Id.of_string "y") g in tclTHENSEQ [ Proofview.V82.of_tactic (Simple.intro id); thin [id]; intros_with_rewrite ] g else if isVar args.(1) && (Environ.evaluable_named (destVar args.(1)) (pf_env g)) then tclTHENSEQ[ unfold_in_concl [(Locus.AllOccurrences, Names.EvalVarRef (destVar args.(1)))]; tclMAP (fun id -> tclTRY(unfold_in_hyp [(Locus.AllOccurrences, Names.EvalVarRef (destVar args.(1)))] ((destVar args.(1)),Locus.InHyp) )) (pf_ids_of_hyps g); intros_with_rewrite ] g else if isVar args.(2) && (Environ.evaluable_named (destVar args.(2)) (pf_env g)) then tclTHENSEQ[ unfold_in_concl [(Locus.AllOccurrences, Names.EvalVarRef (destVar args.(2)))]; tclMAP (fun id -> tclTRY(unfold_in_hyp [(Locus.AllOccurrences, Names.EvalVarRef (destVar args.(2)))] ((destVar args.(2)),Locus.InHyp) )) (pf_ids_of_hyps g); intros_with_rewrite ] g else if isVar args.(1) then let id = pf_get_new_id (Id.of_string "y") g in tclTHENSEQ [ Proofview.V82.of_tactic (Simple.intro id); generalize_dependent_of (destVar args.(1)) id; tclTRY (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar id))); intros_with_rewrite ] g else if isVar args.(2) then let id = pf_get_new_id (Id.of_string "y") g in tclTHENSEQ [ Proofview.V82.of_tactic (Simple.intro id); generalize_dependent_of (destVar args.(2)) id; tclTRY (Proofview.V82.of_tactic (Equality.rewriteRL (mkVar id))); intros_with_rewrite ] g else begin let id = pf_get_new_id (Id.of_string "y") g in tclTHENSEQ[ Proofview.V82.of_tactic (Simple.intro id); tclTRY (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar id))); intros_with_rewrite ] g end | Ind _ when eq_constr t (Coqlib.build_coq_False ()) -> Proofview.V82.of_tactic Tauto.tauto g | Case(_,_,v,_) -> tclTHENSEQ[ Proofview.V82.of_tactic (simplest_case v); intros_with_rewrite ] g | LetIn _ -> tclTHENSEQ[ reduce (Genredexpr.Cbv {Redops.all_flags with Genredexpr.rDelta = false; }) Locusops.onConcl ; intros_with_rewrite ] g | _ -> let id = pf_get_new_id (Id.of_string "y") g in tclTHENSEQ [ Proofview.V82.of_tactic (Simple.intro id);intros_with_rewrite] g end | LetIn _ -> tclTHENSEQ[ reduce (Genredexpr.Cbv {Redops.all_flags with Genredexpr.rDelta = false; }) Locusops.onConcl ; intros_with_rewrite ] g | _ -> tclIDTAC g let rec reflexivity_with_destruct_cases g = let destruct_case () = try match kind_of_term (snd (destApp (pf_concl g))).(2) with | Case(_,_,v,_) -> tclTHENSEQ[ Proofview.V82.of_tactic (simplest_case v); Proofview.V82.of_tactic intros; observe_tac "reflexivity_with_destruct_cases" reflexivity_with_destruct_cases ] | _ -> Proofview.V82.of_tactic reflexivity with e when Errors.noncritical e -> Proofview.V82.of_tactic reflexivity in let eq_ind = make_eq () in let discr_inject = Tacticals.onAllHypsAndConcl ( fun sc g -> match sc with None -> tclIDTAC g | Some id -> match kind_of_term (pf_unsafe_type_of g (mkVar id)) with | App(eq,[|_;t1;t2|]) when eq_constr eq eq_ind -> if Equality.discriminable (pf_env g) (project g) t1 t2 then Proofview.V82.of_tactic (Equality.discrHyp id) g else if Equality.injectable (pf_env g) (project g) t1 t2 then tclTHENSEQ [Proofview.V82.of_tactic (Equality.injHyp None id);thin [id];intros_with_rewrite] g else tclIDTAC g | _ -> tclIDTAC g ) in (tclFIRST [ observe_tac "reflexivity_with_destruct_cases : reflexivity" (Proofview.V82.of_tactic reflexivity); observe_tac "reflexivity_with_destruct_cases : destruct_case" ((destruct_case ())); (* We reach this point ONLY if the same value is matched (at least) two times along binding path. In this case, either we have a discriminable hypothesis and we are done, either at least an injectable one and we do the injection before continuing *) observe_tac "reflexivity_with_destruct_cases : others" (tclTHEN (tclPROGRESS discr_inject ) reflexivity_with_destruct_cases) ]) g (* [prove_fun_complete funs graphs schemes lemmas_types_infos i] is the tactic used to prove completness lemma. [funcs], [graphs] [schemes] [lemmas_types_infos] are the mutually recursive functions (resp. definitions of the graphs of the functions, principles and correctness lemma types) to prove correct. [i] is the indice of the function to prove complete The lemma to prove if suppose to have been generated by [generate_type] (in $\zeta$ normal form that is it looks like~: [\forall (x_1:t_1)\ldots(x_n:t_n), forall res, graph\ x_1\ldots x_n\ res, \rightarrow res = f x_1\ldots x_n in] The sketch of the proof is the following one~: \begin{enumerate} \item intros until $H:graph\ x_1\ldots x_n\ res$ \item $elim\ H$ using schemes.(i) \item for each generated branch, intro the news hyptohesis, for each such hyptohesis [h], if [h] has type [x=?] with [x] a variable, then subst [x], if [h] has type [t=?] with [t] not a variable then rewrite [t] in the subterms, else if [h] is a match then destruct it, else do just introduce it, after all intros, the conclusion should be a reflexive equality. \end{enumerate} *) let prove_fun_complete funcs graphs schemes lemmas_types_infos i : tactic = fun g -> (* We compute the types of the different mutually recursive lemmas in $\zeta$ normal form *) let lemmas = Array.map (fun (_,(ctxt,concl)) -> nf_zeta (Termops.it_mkLambda_or_LetIn concl ctxt)) lemmas_types_infos in (* We get the constant and the principle corresponding to this lemma *) let f = funcs.(i) in let graph_principle = nf_zeta schemes.(i) in let princ_type = pf_unsafe_type_of g graph_principle in let princ_infos = Tactics.compute_elim_sig princ_type in (* Then we get the number of argument of the function and compute a fresh name for each of them *) let nb_fun_args = nb_prod (pf_concl g) - 2 in let args_names = generate_fresh_id (Id.of_string "x") [] nb_fun_args in let ids = args_names@(pf_ids_of_hyps g) in (* and fresh names for res H and the principle (cf bug bug #1174) *) let res,hres,graph_principle_id = match generate_fresh_id (Id.of_string "z") ids 3 with | [res;hres;graph_principle_id] -> res,hres,graph_principle_id | _ -> assert false in let ids = res::hres::graph_principle_id::ids in (* we also compute fresh names for each hyptohesis of each branch of the principle *) let branches = List.rev princ_infos.branches in let intro_pats = List.map (fun (_,_,br_type) -> List.map (fun id -> id) (generate_fresh_id (Id.of_string "y") ids (nb_prod br_type)) ) branches in (* We will need to change the function by its body using [f_equation] if it is recursive (that is the graph is infinite or unfold if the graph is finite *) let rewrite_tac j ids : tactic = let graph_def = graphs.(j) in let infos = try find_Function_infos (fst (destConst funcs.(j))) with Not_found -> error "No graph found" in if infos.is_general || Rtree.is_infinite Declareops.eq_recarg graph_def.mind_recargs then let eq_lemma = try Option.get (infos).equation_lemma with Option.IsNone -> anomaly (Pp.str "Cannot find equation lemma") in tclTHENSEQ[ tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) ids; Proofview.V82.of_tactic (Equality.rewriteLR (mkConst eq_lemma)); (* Don't forget to $\zeta$ normlize the term since the principles have been $\zeta$-normalized *) reduce (Genredexpr.Cbv {Redops.all_flags with Genredexpr.rDelta = false; }) Locusops.onConcl ; Simple.generalize (List.map mkVar ids); thin ids ] else unfold_in_concl [(Locus.AllOccurrences, Names.EvalConstRef (fst (destConst f)))] in (* The proof of each branche itself *) let ind_number = ref 0 in let min_constr_number = ref 0 in let prove_branche i g = (* we fist compute the inductive corresponding to the branch *) let this_ind_number = let constructor_num = i - !min_constr_number in let length = Array.length (graphs.(!ind_number).Declarations.mind_consnames) in if constructor_num <= length then !ind_number else begin incr ind_number; min_constr_number := !min_constr_number + length; !ind_number end in let this_branche_ids = List.nth intro_pats (pred i) in tclTHENSEQ[ (* we expand the definition of the function *) observe_tac "rewrite_tac" (rewrite_tac this_ind_number this_branche_ids); (* introduce hypothesis with some rewrite *) observe_tac "intros_with_rewrite (all)" intros_with_rewrite; (* The proof is (almost) complete *) observe_tac "reflexivity" (reflexivity_with_destruct_cases) ] g in let params_names = fst (List.chop princ_infos.nparams args_names) in let params = List.map mkVar params_names in tclTHENSEQ [ tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) (args_names@[res;hres]); observe_tac "h_generalize" (Simple.generalize [mkApp(applist(graph_principle,params),Array.map (fun c -> applist(c,params)) lemmas)]); Proofview.V82.of_tactic (Simple.intro graph_principle_id); observe_tac "" (tclTHEN_i (observe_tac "elim" (Proofview.V82.of_tactic (elim false None (mkVar hres,NoBindings) (Some (mkVar graph_principle_id,NoBindings))))) (fun i g -> observe_tac "prove_branche" (prove_branche i) g )) ] g (* [derive_correctness make_scheme functional_induction funs graphs] create correctness and completeness lemmas for each function in [funs] w.r.t. [graphs] [make_scheme] is Functional_principle_types.make_scheme (dependency pb) and [functional_induction] is Indfun.functional_induction (same pb) *) let derive_correctness make_scheme functional_induction (funs: pconstant list) (graphs:inductive list) = assert (funs <> []); assert (graphs <> []); let funs = Array.of_list funs and graphs = Array.of_list graphs in let funs_constr = Array.map mkConstU funs in States.with_state_protection_on_exception (fun () -> let env = Global.env () in let evd = ref (Evd.from_env env) in let graphs_constr = Array.map mkInd graphs in let lemmas_types_infos = Util.Array.map2_i (fun i f_constr graph -> (* let const_of_f,u = destConst f_constr in *) let (type_of_lemma_ctxt,type_of_lemma_concl,graph) = generate_type evd false f_constr graph i in let type_info = (type_of_lemma_ctxt,type_of_lemma_concl) in graphs_constr.(i) <- graph; let type_of_lemma = Termops.it_mkProd_or_LetIn type_of_lemma_concl type_of_lemma_ctxt in let _ = Typing.e_type_of (Global.env ()) evd type_of_lemma in let type_of_lemma = nf_zeta type_of_lemma in observe (str "type_of_lemma := " ++ Printer.pr_lconstr_env (Global.env ()) !evd type_of_lemma); type_of_lemma,type_info ) funs_constr graphs_constr in let schemes = (* The functional induction schemes are computed and not saved if there is more that one function if the block contains only one function we can safely reuse [f_rect] *) try if not (Int.equal (Array.length funs_constr) 1) then raise Not_found; [| find_induction_principle evd funs_constr.(0) |] with Not_found -> ( Array.of_list (List.map (fun entry -> (fst (fst(Future.force entry.Entries.const_entry_body)), Option.get entry.Entries.const_entry_type ) ) (make_scheme evd (Array.map_to_list (fun const -> const,GType []) funs)) ) ) in let proving_tac = prove_fun_correct !evd functional_induction funs_constr graphs_constr schemes lemmas_types_infos in Array.iteri (fun i f_as_constant -> let f_id = Label.to_id (con_label (fst f_as_constant)) in (*i The next call to mk_correct_id is valid since we are constructing the lemma Ensures by: obvious i*) let lem_id = mk_correct_id f_id in let (typ,_) = lemmas_types_infos.(i) in Lemmas.start_proof lem_id (Decl_kinds.Global,Flags.is_universe_polymorphism (),((Decl_kinds.Proof Decl_kinds.Theorem))) !evd typ (Lemmas.mk_hook (fun _ _ -> ())); ignore (Pfedit.by (Proofview.V82.tactic (observe_tac ("prove correctness ("^(Id.to_string f_id)^")") (proving_tac i)))); (Lemmas.save_proof (Vernacexpr.(Proved(Transparent,None)))); let finfo = find_Function_infos (fst f_as_constant) in (* let lem_cst = fst (destConst (Constrintern.global_reference lem_id)) in *) let _,lem_cst_constr = Evd.fresh_global (Global.env ()) !evd (Constrintern.locate_reference (Libnames.qualid_of_ident lem_id)) in let (lem_cst,_) = destConst lem_cst_constr in update_Function {finfo with correctness_lemma = Some lem_cst}; ) funs; let lemmas_types_infos = Util.Array.map2_i (fun i f_constr graph -> let (type_of_lemma_ctxt,type_of_lemma_concl,graph) = generate_type evd true f_constr graph i in let type_info = (type_of_lemma_ctxt,type_of_lemma_concl) in graphs_constr.(i) <- graph; let type_of_lemma = Termops.it_mkProd_or_LetIn type_of_lemma_concl type_of_lemma_ctxt in let type_of_lemma = nf_zeta type_of_lemma in observe (str "type_of_lemma := " ++ Printer.pr_lconstr type_of_lemma); type_of_lemma,type_info ) funs_constr graphs_constr in let (kn,_) as graph_ind,u = (destInd graphs_constr.(0)) in let mib,mip = Global.lookup_inductive graph_ind in let sigma, scheme = (Indrec.build_mutual_induction_scheme (Global.env ()) !evd (Array.to_list (Array.mapi (fun i _ -> ((kn,i),u(* Univ.Instance.empty *)),true,InType) mib.Declarations.mind_packets ) ) ) in let schemes = Array.of_list scheme in let proving_tac = prove_fun_complete funs_constr mib.Declarations.mind_packets schemes lemmas_types_infos in Array.iteri (fun i f_as_constant -> let f_id = Label.to_id (con_label (fst f_as_constant)) in (*i The next call to mk_complete_id is valid since we are constructing the lemma Ensures by: obvious i*) let lem_id = mk_complete_id f_id in Lemmas.start_proof lem_id (Decl_kinds.Global,Flags.is_universe_polymorphism (),(Decl_kinds.Proof Decl_kinds.Theorem)) sigma (fst lemmas_types_infos.(i)) (Lemmas.mk_hook (fun _ _ -> ())); ignore (Pfedit.by (Proofview.V82.tactic (observe_tac ("prove completeness ("^(Id.to_string f_id)^")") (proving_tac i)))) ; (Lemmas.save_proof (Vernacexpr.(Proved(Transparent,None)))); let finfo = find_Function_infos (fst f_as_constant) in let _,lem_cst_constr = Evd.fresh_global (Global.env ()) !evd (Constrintern.locate_reference (Libnames.qualid_of_ident lem_id)) in let (lem_cst,_) = destConst lem_cst_constr in update_Function {finfo with completeness_lemma = Some lem_cst} ) funs) () (***********************************************) (* [revert_graph kn post_tac hid] transforme an hypothesis [hid] having type Ind(kn,num) t1 ... tn res when [kn] denotes a graph block into f_num t1... tn = res (by applying [f_complete] to the first type) before apply post_tac on the result if the type of hypothesis has not this form or if we cannot find the completeness lemma then we do nothing *) let revert_graph kn post_tac hid g = let typ = pf_unsafe_type_of g (mkVar hid) in match kind_of_term typ with | App(i,args) when isInd i -> let ((kn',num) as ind'),u = destInd i in if MutInd.equal kn kn' then (* We have generated a graph hypothesis so that we must change it if we can *) let info = try find_Function_of_graph ind' with Not_found -> (* The graphs are mutually recursive but we cannot find one of them !*) anomaly (Pp.str "Cannot retrieve infos about a mutual block") in (* if we can find a completeness lemma for this function then we can come back to the functional form. If not, we do nothing *) match info.completeness_lemma with | None -> tclIDTAC g | Some f_complete -> let f_args,res = Array.chop (Array.length args - 1) args in tclTHENSEQ [ Simple.generalize [applist(mkConst f_complete,(Array.to_list f_args)@[res.(0);mkVar hid])]; thin [hid]; Proofview.V82.of_tactic (Simple.intro hid); post_tac hid ] g else tclIDTAC g | _ -> tclIDTAC g (* [functional_inversion hid fconst f_correct ] is the functional version of [inversion] [hid] is the hypothesis to invert, [fconst] is the function to invert and [f_correct] is the correctness lemma for [fconst]. The sketch is the follwing~: \begin{enumerate} \item Transforms the hypothesis [hid] such that its type is now $res\ =\ f\ t_1 \ldots t_n$ (fails if it is not possible) \item replace [hid] with $R\_f t_1 \ldots t_n res$ using [f_correct] \item apply [inversion] on [hid] \item finally in each branch, replace each hypothesis [R\_f ..] by [f ...] using [f_complete] (whenever such a lemma exists) \end{enumerate} *) let functional_inversion kn hid fconst f_correct : tactic = fun g -> let old_ids = List.fold_right Id.Set.add (pf_ids_of_hyps g) Id.Set.empty in let type_of_h = pf_unsafe_type_of g (mkVar hid) in match kind_of_term type_of_h with | App(eq,args) when eq_constr eq (make_eq ()) -> let pre_tac,f_args,res = match kind_of_term args.(1),kind_of_term args.(2) with | App(f,f_args),_ when eq_constr f fconst -> ((fun hid -> Proofview.V82.of_tactic (intros_symmetry (Locusops.onHyp hid))),f_args,args.(2)) |_,App(f,f_args) when eq_constr f fconst -> ((fun hid -> tclIDTAC),f_args,args.(1)) | _ -> (fun hid -> tclFAIL 1 (mt ())),[||],args.(2) in tclTHENSEQ[ pre_tac hid; Simple.generalize [applist(f_correct,(Array.to_list f_args)@[res;mkVar hid])]; thin [hid]; Proofview.V82.of_tactic (Simple.intro hid); Proofview.V82.of_tactic (Inv.inv FullInversion None (NamedHyp hid)); (fun g -> let new_ids = List.filter (fun id -> not (Id.Set.mem id old_ids)) (pf_ids_of_hyps g) in tclMAP (revert_graph kn pre_tac) (hid::new_ids) g ); ] g | _ -> tclFAIL 1 (mt ()) g let invfun qhyp f = let f = match f with | ConstRef f -> f | _ -> raise (Errors.UserError("",str "Not a function")) in try let finfos = find_Function_infos f in let f_correct = mkConst(Option.get finfos.correctness_lemma) and kn = fst finfos.graph_ind in Proofview.V82.of_tactic ( Tactics.try_intros_until (fun hid -> Proofview.V82.tactic (functional_inversion kn hid (mkConst f) f_correct)) qhyp ) with | Not_found -> error "No graph found" | Option.IsNone -> error "Cannot use equivalence with graph!" let invfun qhyp f g = match f with | Some f -> invfun qhyp f g | None -> Proofview.V82.of_tactic begin Tactics.try_intros_until (fun hid -> Proofview.V82.tactic begin fun g -> let hyp_typ = pf_unsafe_type_of g (mkVar hid) in match kind_of_term hyp_typ with | App(eq,args) when eq_constr eq (make_eq ()) -> begin let f1,_ = decompose_app args.(1) in try if not (isConst f1) then failwith ""; let finfos = find_Function_infos (fst (destConst f1)) in let f_correct = mkConst(Option.get finfos.correctness_lemma) and kn = fst finfos.graph_ind in functional_inversion kn hid f1 f_correct g with | Failure "" | Option.IsNone | Not_found -> try let f2,_ = decompose_app args.(2) in if not (isConst f2) then failwith ""; let finfos = find_Function_infos (fst (destConst f2)) in let f_correct = mkConst(Option.get finfos.correctness_lemma) and kn = fst finfos.graph_ind in functional_inversion kn hid f2 f_correct g with | Failure "" -> errorlabstrm "" (str "Hypothesis " ++ Ppconstr.pr_id hid ++ str " must contain at least one Function") | Option.IsNone -> if do_observe () then error "Cannot use equivalence with graph for any side of the equality" else errorlabstrm "" (str "Cannot find inversion information for hypothesis " ++ Ppconstr.pr_id hid) | Not_found -> if do_observe () then error "No graph found for any side of equality" else errorlabstrm "" (str "Cannot find inversion information for hypothesis " ++ Ppconstr.pr_id hid) end | _ -> errorlabstrm "" (Ppconstr.pr_id hid ++ str " must be an equality ") end) qhyp end g