|
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
open Ltac_plugin
open Declarations
open CErrors
open Util
open Names
open Term
open Constr
open Context
open EConstr
open Vars
open Pp
open Globnames
open Tacticals
open Tactics
open Indfun_common
open Tacmach
open Tactypes
open Termops
open Context.Rel.Declaration
module RelDecl = Context.Rel.Declaration
(* The local debugging mechanism *)
(* let msgnl = Pp.msgnl *)
let observe strm =
if do_observe ()
then Feedback.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 CErrors.noncritical e -> assert false
in
try
let v = tac g in
msgnl (goal ++ fnl () ++ s ++(str " ")++(str "finished")); v
with reraise ->
let reraise = CErrors.push reraise in
let e = ExplainErr.process_vernac_interp_error reraise in
observe (hov 0 (str "observation "++ s++str " raised exception " ++
CErrors.iprint e ++ str " on goal" ++ fnl() ++ goal ));
iraise reraise;;
let observe_tac s tac g =
if do_observe ()
then do_observe_tac (str s) tac g
else tac g
let thin ids gl = Proofview.V82.of_tactic (Tactics.clear ids) gl
(* (\* [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
EConstr.of_constr (UnivGen.constr_of_monomorphic_global (Coqlib.lib_ref "core.eq.type"))
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 !evd graph)))
in
evd:=evd';
let sigma, graph_arity = Typing.type_of (Global.env ()) !evd graph in
evd := sigma;
let ctxt,_ = decompose_prod_assum !evd graph_arity in
let fun_ctxt,res_type =
match ctxt with
| [] | [_] -> anomaly (Pp.str "Not a valid context.")
| decl :: fun_ctxt -> fun_ctxt, RelDecl.get_type decl
in
let rec args_from_decl i accu = function
| [] -> accu
| LocalDef _ :: 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 = fun decl -> match RelDecl.get_name decl with
| Name id -> Some id
| Anonymous -> None
in
let named_ctxt = Id.Set.of_list (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") (Id.Set.add 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 =
LocalAssum (make_annot (Name res_id) Sorts.Relevant, lift 1 res_type) ::
LocalDef (make_annot (Name fv_id) Sorts.Relevant, 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 LocalAssum (make_annot Anonymous Sorts.Relevant,graph_applied)::pre_ctxt,(lift 1 res_eq_f_of_args),graph
else LocalAssum (make_annot Anonymous Sorts.Relevant,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 EConstr.kind !evd f with
| Const c' -> c'
| _ -> user_err Pp.(str "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 (Id.Set.of_list avoid) in
id::(generate_fresh_id x (id::avoid) (pred i))
(* [prove_fun_correct funs_constr graphs_constr schemes lemmas_types_infos i ]
is the tactic used to prove correctness lemma.
[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 funs_constr graphs_constr schemes lemmas_types_infos i : Tacmach.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 evd 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 = Reductionops.nf_zeta (Global.env ()) evd princ_type in
let princ_infos = Tactics.compute_elim_sig evd princ_type in
(* The number of args of the function is then easily computable *)
let nb_fun_args = nb_prod (project g) (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") (Id.Set.of_list 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 decl ->
List.map
(fun id -> CAst.make @@ IntroNaming (Namegen.IntroIdentifier id))
(generate_fresh_id (Id.of_string "y") ids (List.length (fst (decompose_prod_assum evd (RelDecl.get_type decl)))))
)
branches
in
(* before building the full intro pattern for the principle *)
let eq_ind = make_eq () in
let eq_construct = mkConstructUi (destInd evd 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 {CAst.v=pat} acc ->
match pat with
| IntroNaming (Namegen.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
let sigma = project g in
match EConstr.kind sigma type_of_hid with
| Prod(_,_,t') ->
begin
match EConstr.kind sigma t' with
| Prod(_,t'',t''') ->
begin
match EConstr.kind sigma t'',EConstr.kind sigma t''' with
| App(eq,args), App(graph',_)
when
(EConstr.eq_constr sigma eq eq_ind) &&
Array.exists (EConstr.eq_constr_nounivs sigma 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); *)
(
tclTHENLIST
[
observe_tac("h_intro_patterns ") (let l = (List.nth intro_pats (pred i)) in
match l with
| [] -> tclIDTAC
| _ -> Proofview.V82.of_tactic (intro_patterns false l));
(* unfolding of all the defined variables introduced by this branch *)
(* observe_tac "unfolding" pre_tac; *)
(* $zeta$ normalizing of the conclusion *)
Proofview.V82.of_tactic (reduce
(Genredexpr.Cbv
{ Redops.all_flags with
Genredexpr.rDelta = false ;
Genredexpr.rConst = []
}
)
Locusops.onConcl);
observe_tac ("toto ") tclIDTAC;
(* introducing 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::decl::ctxt ->
let res = EConstr.it_mkLambda_or_LetIn
(EConstr.it_mkProd_or_LetIn concl [hres;res])
(LocalAssum (RelDecl.get_annot decl, RelDecl.get_type decl) :: 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) decl p ->
let id = Namegen.next_ident_away (Nameops.Name.get_id (RelDecl.get_name decl)) (Id.Set.of_list 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) decl p ->
let id = Namegen.next_ident_away (Nameops.Name.get_id (RelDecl.get_name decl)) (Id.Set.of_list avoid) in
(Reductionops.nf_zeta (pf_env g) (project g) p)::bindings,id::avoid)
([],avoid)
princ_infos.predicates
(lemmas)))
in
(params_bindings@lemmas_bindings)
in
tclTHENLIST
[
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 =
let open Context.Named.Declaration in
tclMAP
(function
| LocalAssum ({binder_name=id},t) when not (Id.equal id hyp) &&
(Termops.occur_var (pf_env g) (project g) x t) -> tclTHEN (Proofview.V82.of_tactic (Tactics.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 tauto =
let dp = List.map Id.of_string ["Tauto" ; "Init"; "Coq"] in
let mp = ModPath.MPfile (DirPath.make dp) in
let kn = KerName.make mp (Label.make "tauto") in
Proofview.tclBIND (Proofview.tclUNIT ()) begin fun () ->
let body = Tacenv.interp_ltac kn in
Tacinterp.eval_tactic body
end
let rec intros_with_rewrite g =
observe_tac "intros_with_rewrite" intros_with_rewrite_aux g
and intros_with_rewrite_aux : Tacmach.tactic =
fun g ->
let eq_ind = make_eq () in
let sigma = project g in
match EConstr.kind sigma (pf_concl g) with
| Prod(_,t,t') ->
begin
match EConstr.kind sigma t with
| App(eq,args) when (EConstr.eq_constr sigma 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
tclTHENLIST [ Proofview.V82.of_tactic (Simple.intro id); thin [id]; intros_with_rewrite ] g
else if isVar sigma args.(1) && (Environ.evaluable_named (destVar sigma args.(1)) (pf_env g))
then tclTHENLIST[
Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, Names.EvalVarRef (destVar sigma args.(1)))]);
tclMAP (fun id -> tclTRY(Proofview.V82.of_tactic (unfold_in_hyp [(Locus.AllOccurrences, Names.EvalVarRef (destVar sigma args.(1)))] ((destVar sigma args.(1)),Locus.InHyp) )))
(pf_ids_of_hyps g);
intros_with_rewrite
] g
else if isVar sigma args.(2) && (Environ.evaluable_named (destVar sigma args.(2)) (pf_env g))
then tclTHENLIST[
Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, Names.EvalVarRef (destVar sigma args.(2)))]);
tclMAP (fun id -> tclTRY(Proofview.V82.of_tactic (unfold_in_hyp [(Locus.AllOccurrences, Names.EvalVarRef (destVar sigma args.(2)))] ((destVar sigma args.(2)),Locus.InHyp) )))
(pf_ids_of_hyps g);
intros_with_rewrite
] g
else if isVar sigma args.(1)
then
let id = pf_get_new_id (Id.of_string "y") g in
tclTHENLIST [ Proofview.V82.of_tactic (Simple.intro id);
generalize_dependent_of (destVar sigma args.(1)) id;
tclTRY (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar id)));
intros_with_rewrite
]
g
else if isVar sigma args.(2)
then
let id = pf_get_new_id (Id.of_string "y") g in
tclTHENLIST [ Proofview.V82.of_tactic (Simple.intro id);
generalize_dependent_of (destVar sigma 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
tclTHENLIST[
Proofview.V82.of_tactic (Simple.intro id);
tclTRY (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar id)));
intros_with_rewrite
] g
end
| Ind _ when EConstr.eq_constr sigma t (EConstr.of_constr (UnivGen.constr_of_monomorphic_global @@ Coqlib.lib_ref "core.False.type")) ->
Proofview.V82.of_tactic tauto g
| Case(_,_,v,_) ->
tclTHENLIST[
Proofview.V82.of_tactic (simplest_case v);
intros_with_rewrite
] g
| LetIn _ ->
tclTHENLIST[
Proofview.V82.of_tactic (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
tclTHENLIST [ Proofview.V82.of_tactic (Simple.intro id);intros_with_rewrite] g
end
| LetIn _ ->
tclTHENLIST[
Proofview.V82.of_tactic (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 EConstr.kind (project g) (snd (destApp (project g) (pf_concl g))).(2) with
| Case(_,_,v,_) ->
tclTHENLIST[
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 CErrors.noncritical e -> Proofview.V82.of_tactic reflexivity
in
let eq_ind = make_eq () in
let my_inj_flags = Some {
Equality.keep_proof_equalities = false;
injection_in_context = false; (* for compatibility, necessary *)
injection_pattern_l2r_order = false; (* probably does not matter; except maybe with dependent hyps *)
} in
let discr_inject =
Tacticals.onAllHypsAndConcl (
fun sc g ->
match sc with
None -> tclIDTAC g
| Some id ->
match EConstr.kind (project g) (pf_unsafe_type_of g (mkVar id)) with
| App(eq,[|_;t1;t2|]) when EConstr.eq_constr (project g) 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) ~keep_proofs:None t1 t2
then tclTHENLIST [Proofview.V82.of_tactic (Equality.injHyp my_inj_flags 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 completeness 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 : Tacmach.tactic =
fun g ->
(* We compute the types of the different mutually recursive lemmas
in $\zeta$ normal form
*)
let lemmas =
Array.map
(fun (_,(ctxt,concl)) -> Reductionops.nf_zeta (pf_env g) (project g) (EConstr.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 = Reductionops.nf_zeta (pf_env g) (project g) (EConstr.of_constr schemes.(i)) in
let princ_type = pf_unsafe_type_of g graph_principle in
let princ_infos = Tactics.compute_elim_sig (project g) 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 (project g) (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 decl ->
List.map
(fun id -> id)
(generate_fresh_id (Id.of_string "y") ids (nb_prod (project g) (RelDecl.get_type decl)))
)
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 : Tacmach.tactic =
let graph_def = graphs.(j) in
let infos =
try find_Function_infos (fst (destConst (project g) funcs.(j)))
with Not_found -> user_err Pp.(str "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
tclTHENLIST[
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 *)
Proofview.V82.of_tactic (reduce
(Genredexpr.Cbv
{Redops.all_flags
with Genredexpr.rDelta = false;
})
Locusops.onConcl)
;
Proofview.V82.of_tactic (generalize (List.map mkVar ids));
thin ids
]
else
Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, Names.EvalConstRef (fst (destConst (project g) 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
tclTHENLIST[
(* 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 open EConstr in
let params = List.map mkVar params_names in
tclTHENLIST
[ tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) (args_names@[res;hres]);
observe_tac "h_generalize"
(Proofview.V82.of_tactic (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 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
*)
let derive_correctness make_scheme (funs: pconstant list) (graphs:inductive list) =
assert (funs <> []);
assert (graphs <> []);
let funs = Array.of_list funs and graphs = Array.of_list graphs in
let map (c, u) = mkConstU (c, EInstance.make u) in
let funs_constr = Array.map map funs in
(* XXX STATE Why do we need this... why is the toplevel protection not enough *)
funind_purify
(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 = EConstr.it_mkProd_or_LetIn type_of_lemma_concl type_of_lemma_ctxt in
let sigma, _ = Typing.type_of (Global.env ()) !evd type_of_lemma in
evd := sigma;
let type_of_lemma = Reductionops.nf_zeta (Global.env ()) !evd type_of_lemma in
observe (str "type_of_lemma := " ++ Printer.pr_leconstr_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 ->
(EConstr.of_constr (fst (fst(Future.force entry.Entries.const_entry_body))), EConstr.of_constr (Option.get entry.Entries.const_entry_type ))
)
(make_scheme evd (Array.map_to_list (fun const -> const,Sorts.InType) funs))
)
)
in
let proving_tac =
prove_fun_correct !evd funs_constr graphs_constr schemes lemmas_types_infos
in
Array.iteri
(fun i f_as_constant ->
let f_id = Label.to_id (Constant.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
let pstate = Lemmas.start_proof ~ontop:None
lem_id
(Decl_kinds.Global,false,((Decl_kinds.Proof Decl_kinds.Theorem)))
!evd
typ in
let pstate = fst @@ Pfedit.by
(Proofview.V82.tactic (observe_tac ("prove correctness ("^(Id.to_string f_id)^")")
(proving_tac i))) pstate in
let _ = Lemmas.save_proof_proved ?proof:None ~pstate ~opaque:Proof_global.Transparent ~idopt:None in
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 !evd 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 =
EConstr.it_mkProd_or_LetIn type_of_lemma_concl type_of_lemma_ctxt
in
let type_of_lemma = Reductionops.nf_zeta env !evd type_of_lemma in
observe (str "type_of_lemma := " ++ Printer.pr_leconstr_env env !evd type_of_lemma);
type_of_lemma,type_info
)
funs_constr
graphs_constr
in
let (kn,_) as graph_ind,u = (destInd !evd 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), EInstance.kind !evd u),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 (Constant.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
let pstate = Lemmas.start_proof ~ontop:None lem_id
(Decl_kinds.Global,false,(Decl_kinds.Proof Decl_kinds.Theorem)) sigma
(fst lemmas_types_infos.(i)) in
let pstate = fst (Pfedit.by
(Proofview.V82.tactic (observe_tac ("prove completeness ("^(Id.to_string f_id)^")")
(proving_tac i))) pstate) in
let _pstate = Lemmas.save_proof_proved ?proof:None ~pstate ~opaque:Proof_global.Transparent ~idopt:None in
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 !evd 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 sigma = project g in
let typ = pf_unsafe_type_of g (mkVar hid) in
match EConstr.kind sigma typ with
| App(i,args) when isInd sigma i ->
let ((kn',num) as ind'),u = destInd sigma 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
tclTHENLIST
[
Proofview.V82.of_tactic (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 following~:
\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 : Tacmach.tactic =
fun g ->
let old_ids = List.fold_right Id.Set.add (pf_ids_of_hyps g) Id.Set.empty in
let sigma = project g in
let type_of_h = pf_unsafe_type_of g (mkVar hid) in
match EConstr.kind sigma type_of_h with
| App(eq,args) when EConstr.eq_constr sigma eq (make_eq ()) ->
let pre_tac,f_args,res =
match EConstr.kind sigma args.(1),EConstr.kind sigma args.(2) with
| App(f,f_args),_ when EConstr.eq_constr sigma f fconst ->
((fun hid -> Proofview.V82.of_tactic (intros_symmetry (Locusops.onHyp hid))),f_args,args.(2))
|_,App(f,f_args) when EConstr.eq_constr sigma f fconst ->
((fun hid -> tclIDTAC),f_args,args.(1))
| _ -> (fun hid -> tclFAIL 1 (mt ())),[||],args.(2)
in
tclTHENLIST [
pre_tac hid;
Proofview.V82.of_tactic (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 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 error msg = user_err Pp.(str msg)
let invfun qhyp f =
let f =
match f with
| ConstRef f -> f
| _ -> raise (CErrors.UserError(None,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!"
exception NoFunction
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 sigma = project g in
let hyp_typ = pf_unsafe_type_of g (mkVar hid) in
match EConstr.kind sigma hyp_typ with
| App(eq,args) when EConstr.eq_constr sigma eq (make_eq ()) ->
begin
let f1,_ = decompose_app sigma args.(1) in
try
if not (isConst sigma f1) then raise NoFunction;
let finfos = find_Function_infos (fst (destConst sigma 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 | NoFunction | Option.IsNone | Not_found ->
try
let f2,_ = decompose_app sigma args.(2) in
if not (isConst sigma f2) then raise NoFunction;
let finfos = find_Function_infos (fst (destConst sigma 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
| NoFunction ->
user_err (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 user_err (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 user_err (str "Cannot find inversion information for hypothesis " ++ Ppconstr.pr_id hid)
end
| _ -> user_err (Ppconstr.pr_id hid ++ str " must be an equality ")
end)
qhyp
end
g
[ Dauer der Verarbeitung: 0.15 Sekunden
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