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Datei: module_with_def_univ_poly.v Sprache: SML

Original von: Coq^©

open Printer
open CErrors
open Util
open Constr
open Context
open EConstr
open Vars
open Namegen
open Names
open Pp
open Tacmach
open Termops
open Tacticals
open Tactics
open Indfun_common
open Libnames
open Globnames
open Context.Rel.Declaration

module RelDecl = Context.Rel.Declaration

(* let msgnl = Pp.msgnl *)

(*
let observe strm =
  if do_observe ()
  then Pp.msg_debug strm
  else ()

let do_observe_tac s tac g =
try let v = tac g in (* msgnl (goal ++ fnl () ++ (str s)++(str " ")++(str "finished")); *)
with e ->
   let e = ExplainErr.process_vernac_interp_error e in
   let goal = begin try (Printer.pr_goal g) with _ -> assert false end in
   msg_debug (str "observation "++ s++str " raised exception " ++
     Errors.print e ++ str " on goal " ++ goal );
   raise e;;

let observe_tac_stream s tac g =
  if do_observe ()
  then do_observe_tac  s tac g
  else tac g

let observe_tac s tac g = observe_tac_stream (str s) tac g
  *)

let debug_queue = Stack.create ()

let rec print_debug_queue e =
  if  not (Stack.is_empty debug_queue)
  then
    begin
      let lmsg,goal = Stack.pop debug_queue in
      let _ =
match e with
| Some e ->
    Feedback.msg_debug (hov 0 (lmsg ++ (str " raised exception " ++ CErrors.print e) ++ str " on goal" ++ fnl() ++ goal))
| None ->
    begin
      Feedback.msg_debug (str " from " ++ lmsg ++ str " on goal" ++ fnl() ++ goal);
    end in
      print_debug_queue None ;
    end

let observe strm =
  if do_observe ()
  then Feedback.msg_debug strm
  else ()

let do_observe_tac s tac g =
  let goal = Printer.pr_goal g in
  let lmsg = (str "observation : ") ++ s in
  Stack.push (lmsg,goal) debug_queue;
  try
    let v = tac g in
    ignore(Stack.pop debug_queue);
    v
  with reraise ->
    let reraise = CErrors.push reraise in
    if not (Stack.is_empty debug_queue)
    then print_debug_queue (Some (fst (ExplainErr.process_vernac_interp_error reraise)));
    iraise reraise

let observe_tac_stream s tac g =
  if do_observe ()
  then do_observe_tac s tac g
  else tac g

let observe_tac s = observe_tac_stream (str s)


let list_chop ?(msg="") n l =
  try
    List.chop n l
  with Failure (msg') ->
    failwith (msg ^ msg')

let pop t = Vars.lift (-1) t

let make_refl_eq constructor type_of_t t  =
(*   let refl_equal_term = Lazy.force refl_equal in *)
  mkApp(constructor,[|type_of_t;t|])

type pte_info =
    {
      proving_tac : (Id.t list -> Tacmach.tactic);
      is_valid : constr -> bool
    }

type ptes_info = pte_info Id.Map.t

type 'a dynamic_info =
    {
      nb_rec_hyps : int;
      rec_hyps : Id.t list ;
      eq_hyps : Id.t list;
      info : 'a
    }

type body_info = constr dynamic_info

let finish_proof dynamic_infos g =
  observe_tac "finish"
    (Proofview.V82.of_tactic assumption)
    g

let refine c = Refiner.refiner ~check:true EConstr.Unsafe.(to_constr c)

let thin l = Proofview.V82.of_tactic (Tactics.clear l)

let eq_constr sigma u v = EConstr.eq_constr_nounivs sigma u v

let is_trivial_eq sigma t =
  let res =   try
    begin
      match EConstr.kind sigma t with
| App(f,[|_;t1;t2|]) when eq_constr sigma f (Lazy.force eq) ->
     eq_constr sigma t1 t2
| App(f,[|t1;a1;t2;a2|]) when eq_constr sigma f (jmeq ())  ->
     eq_constr sigma t1 t2 && eq_constr sigma a1 a2
| _ -> false
    end
  with e when CErrors.noncritical e -> false
  in
(*   observe (str "is_trivial_eq " ++ Printer.pr_lconstr t ++ (if res then str " true" else str " false")); *)
  res

let rec incompatible_constructor_terms sigma t1 t2 =
  let c1,arg1 = decompose_app sigma t1
  and c2,arg2 = decompose_app sigma t2
  in
  (not (eq_constr sigma t1 t2)) &&
    isConstruct sigma c1 && isConstruct sigma c2 &&
    (
      not (eq_constr sigma c1 c2) ||
List.exists2 (incompatible_constructor_terms sigma) arg1 arg2
    )

let is_incompatible_eq env sigma t =
  let res =
    try
      match EConstr.kind sigma t with
| App(f,[|_;t1;t2|]) when eq_constr sigma f (Lazy.force eq) ->
     incompatible_constructor_terms sigma t1 t2
| App(f,[|u1;t1;u2;t2|]) when eq_constr sigma f (jmeq ()) ->
     (eq_constr sigma u1 u2 &&
        incompatible_constructor_terms sigma t1 t2)
| _ -> false
    with e when CErrors.noncritical e -> false
  in
  if res then   observe (str "is_incompatible_eq " ++ pr_leconstr_env env sigma t);
  res

let change_hyp_with_using msg hyp_id t tac : tactic =
  fun g ->
    let prov_id = pf_get_new_id hyp_id g in
    tclTHENS
      ((* observe_tac msg *) Proofview.V82.of_tactic (assert_by (Name prov_id) t (Proofview.V82.tactic (tclCOMPLETE tac))))
      [tclTHENLIST
      [
(* observe_tac "change_hyp_with_using thin" *) (thin [hyp_id]);
(* observe_tac "change_hyp_with_using rename " *) (Proofview.V82.of_tactic (rename_hyp [prov_id,hyp_id]))
      ]] g

exception TOREMOVE

let prove_trivial_eq h_id context (constructor,type_of_term,term) =
  let nb_intros = List.length context in
  tclTHENLIST
    [
      tclDO nb_intros (Proofview.V82.of_tactic intro); (* introducing context *)
      (fun g ->
  let context_hyps =
    fst (list_chop ~msg:"prove_trivial_eq : " nb_intros (pf_ids_of_hyps g))
  in
  let context_hyps' =
    (mkApp(constructor,[|type_of_term;term|]))::
      (List.map mkVar context_hyps)
  in
  let to_refine = applist(mkVar h_id,List.rev context_hyps') in
  refine to_refine g
      )
    ]

let find_rectype env sigma c =
  let (t, l) = decompose_app sigma (Reductionops.whd_betaiotazeta sigma c) in
  match EConstr.kind sigma t with
  | Ind ind -> (t, l)
  | Construct _ -> (t,l)
  | _ -> raise Not_found

let isAppConstruct ?(env=Global.env ()) sigma t =
  try
    let t',l = find_rectype env sigma t in
    observe (str "isAppConstruct : " ++ Printer.pr_leconstr_env env sigma t ++ str " -> " ++
             Printer.pr_leconstr_env env sigma (applist  (t',l)));
    true
  with Not_found -> false

exception NoChange

let change_eq env sigma hyp_id (context:rel_context) x t end_of_type  =
  let nochange ?t' msg =
    begin
      observe (str ("Not treating ( "^msg^" )") ++ pr_leconstr_env env sigma t  ++ str " " ++
               match t' with None -> str "" | Some t -> Printer.pr_leconstr_env env sigma t );
      raise NoChange;
    end
  in
  let eq_constr c1 c2 =
    try ignore(Evarconv.unify_delay env sigma c1 c2); true
    with Evarconv.UnableToUnify _ -> false in
  if not (noccurn sigma 1 end_of_type)
  then nochange "dependent"; (* if end_of_type depends on this term we don't touch it  *)
    if not (isApp sigma t) then nochange "not an equality";
    let f_eq,args = destApp sigma t in
    let constructor,t1,t2,t1_typ =
      try
if (eq_constr f_eq (Lazy.force eq))
then
   let t1 = (args.(1),args.(0))
   and t2 = (args.(2),args.(0))
   and t1_typ = args.(0)
   in
   (Lazy.force refl_equal,t1,t2,t1_typ)
else
   if (eq_constr f_eq (jmeq ()))
   then
     (jmeq_refl (),(args.(1),args.(0)),(args.(3),args.(2)),args.(0))
   else nochange "not an equality"
      with e when CErrors.noncritical e -> nochange "not an equality"
    in
    if not ((closed0 sigma (fst t1)) && (closed0 sigma (snd t1)))then nochange "not a closed lhs";
    let rec compute_substitution sub t1 t2 =
(*       observe (str "compute_substitution : " ++ pr_lconstr t1 ++ str " === " ++ pr_lconstr t2); *)
      if isRel sigma t2
      then
let t2 = destRel sigma t2  in
begin
   try
     let t1' = Int.Map.find t2 sub in
     if not (eq_constr t1 t1') then nochange "twice bound variable";
     sub
   with Not_found ->
     assert (closed0 sigma t1);
     Int.Map.add t2 t1 sub
end
      else if isAppConstruct sigma t1 && isAppConstruct sigma t2
      then
begin
   let c1,args1 =  find_rectype env sigma t1
   and c2,args2 = find_rectype env sigma t2
   in
   if not (eq_constr c1 c2) then nochange "cannot solve (diff)";
   List.fold_left2 compute_substitution sub args1 args2
end
      else
if (eq_constr t1 t2) then sub else nochange ~t':(make_refl_eq constructor (Reductionops.whd_all env sigma t1) t2) "cannot solve (diff)"
    in
    let sub = compute_substitution Int.Map.empty (snd t1) (snd t2) in
    let sub = compute_substitution sub (fst t1) (fst t2) in
    let end_of_type_with_pop = pop end_of_type in (*the equation will be removed *)
    let new_end_of_type =
      (* Ugly hack to prevent Map.fold order change between ocaml-3.08.3 and ocaml-3.08.4
Can be safely replaced by the next comment for Ocaml >= 3.08.4
      *)
      let sub = Int.Map.bindings sub in
      List.fold_left (fun end_of_type (i,t)  -> liftn 1 i (substnl  [t] (i-1) end_of_type))
end_of_type_with_pop
sub
    in
    let old_context_length = List.length context + 1 in
    let witness_fun =
      mkLetIn(make_annot Anonymous Sorts.Relevant,make_refl_eq constructor t1_typ (fst t1),t,
        mkApp(mkVar hyp_id,Array.init old_context_length (fun i -> mkRel (old_context_length - i)))
       )
    in
    let new_type_of_hyp,ctxt_size,witness_fun =
      List.fold_left_i
(fun i (end_of_type,ctxt_size,witness_fun) decl ->
    try
      let witness = Int.Map.find i sub in
      if is_local_def decl then anomaly (Pp.str "can not redefine a rel!");
      (pop end_of_type,ctxt_size,mkLetIn (RelDecl.get_annot decl,
                                            witness, RelDecl.get_type decl, witness_fun))
    with Not_found  ->
      (mkProd_or_LetIn decl end_of_type, ctxt_size + 1, mkLambda_or_LetIn decl witness_fun)
)
1
(new_end_of_type,0,witness_fun)
context
    in
    let new_type_of_hyp =
      Reductionops.nf_betaiota env sigma new_type_of_hyp in
    let new_ctxt,new_end_of_type =
      decompose_prod_n_assum sigma ctxt_size new_type_of_hyp
    in
    let prove_new_hyp : tactic =
      tclTHEN
(tclDO ctxt_size (Proofview.V82.of_tactic intro))
(fun g ->
    let all_ids = pf_ids_of_hyps g in
    let new_ids,_  = list_chop ctxt_size all_ids in
    let to_refine = applist(witness_fun,List.rev_map mkVar new_ids) in
    let evm, _ = pf_apply Typing.type_of g to_refine in
      tclTHEN (Refiner.tclEVARS evm) (refine to_refine) g
)
    in
    let simpl_eq_tac =
      change_hyp_with_using "prove_pattern_simplification" hyp_id new_type_of_hyp prove_new_hyp
    in
(*     observe (str "In " ++ Ppconstr.pr_id hyp_id ++  *)
(*        str "removing an equation " ++ fnl ()++  *)
(*        str "old_typ_of_hyp :=" ++ *)
(*        Printer.pr_lconstr_env *)
(*        env *)
(*        (it_mkProd_or_LetIn ~init:end_of_type ((x,None,t)::context)) *)
(*      ++ fnl () ++ *)
(*        str "new_typ_of_hyp := "++  *)
(*        Printer.pr_lconstr_env env new_type_of_hyp ++ fnl () *)
(*      ++ str "old context := " ++ pr_rel_context env context ++ fnl ()  *)
(*      ++ str "new context := " ++ pr_rel_context env new_ctxt ++ fnl ()  *)
(*      ++ str "old type  := " ++ pr_lconstr end_of_type ++ fnl ()  *)
(*      ++ str "new type := " ++ pr_lconstr new_end_of_type ++ fnl ()  *)
(* ); *)
    new_ctxt,new_end_of_type,simpl_eq_tac

let is_property sigma (ptes_info:ptes_info) t_x full_type_of_hyp =
  if isApp sigma t_x
  then
    let pte,args = destApp sigma t_x in
    if isVar sigma pte && Array.for_all (closed0 sigma) args
    then
      try
let info = Id.Map.find (destVar sigma pte) ptes_info in
info.is_valid full_type_of_hyp
      with Not_found -> false
    else false
  else false

let isLetIn sigma t =
  match EConstr.kind sigma t with
    | LetIn _ -> true
    | _ -> false

let h_reduce_with_zeta cl =
  Proofview.V82.of_tactic (reduce
    (Genredexpr.Cbv
       {Redops.all_flags
with Genredexpr.rDelta = false;
       }) cl)

let rewrite_until_var arg_num eq_ids : tactic =
  (* tests if the declares recursive argument is neither a Constructor nor
     an applied Constructor since such a form for the recursive argument
     will break the Guard when trying to save the Lemma.
  *)
  let test_var g =
    let sigma = project g in
    let _,args = destApp sigma (pf_concl g) in
    not ((isConstruct sigma args.(arg_num)) || isAppConstruct sigma args.(arg_num))
  in
  let rec do_rewrite eq_ids g  =
    if test_var g
    then tclIDTAC g
    else
      match eq_ids with
| [] -> anomaly (Pp.str "Cannot find a way to prove recursive property.");
| eq_id::eq_ids ->
     tclTHEN
       (tclTRY (Proofview.V82.of_tactic (Equality.rewriteRL (mkVar eq_id))))
       (do_rewrite eq_ids)
       g
  in
  do_rewrite eq_ids

let rec_pte_id = Id.of_string "Hrec"
let clean_hyp_with_heq ptes_infos eq_hyps hyp_id env sigma =
  let coq_False = EConstr.of_constr (UnivGen.constr_of_monomorphic_global @@ Coqlib.lib_ref "core.False.type") in
  let coq_True = EConstr.of_constr (UnivGen.constr_of_monomorphic_global @@ Coqlib.lib_ref "core.True.type") in
  let coq_I = EConstr.of_constr (UnivGen.constr_of_monomorphic_global @@ Coqlib.lib_ref "core.True.I") in
  let rec scan_type  context type_of_hyp : tactic =
    if isLetIn sigma type_of_hyp then
      let real_type_of_hyp = it_mkProd_or_LetIn type_of_hyp context in
      let reduced_type_of_hyp = Reductionops.nf_betaiotazeta env sigma real_type_of_hyp in
      (* length of context didn't change ? *)
      let new_context,new_typ_of_hyp =
  decompose_prod_n_assum sigma (List.length context) reduced_type_of_hyp
      in
        tclTHENLIST
[ h_reduce_with_zeta (Locusops.onHyp hyp_id);
   scan_type new_context new_typ_of_hyp ]
    else if isProd sigma type_of_hyp
    then
      begin
        let (x,t_x,t') = destProd sigma type_of_hyp in
let actual_real_type_of_hyp = it_mkProd_or_LetIn t' context in
if is_property sigma ptes_infos t_x actual_real_type_of_hyp then
   begin
     let pte,pte_args =  (destApp sigma t_x) in
     let (* fix_info *) prove_rec_hyp = (Id.Map.find (destVar sigma pte) ptes_infos).proving_tac in
     let popped_t' = pop t' in
     let real_type_of_hyp = it_mkProd_or_LetIn popped_t' context in
     let prove_new_type_of_hyp =
       let context_length = List.length context in
       tclTHENLIST
  [
    tclDO context_length (Proofview.V82.of_tactic intro);
    (fun g ->
       let context_hyps_ids =
         fst (list_chop ~msg:"rec hyp : context_hyps"
         context_length (pf_ids_of_hyps g))
       in
       let rec_pte_id = pf_get_new_id rec_pte_id g in
       let to_refine =
         applist(mkVar hyp_id,
          List.rev_map mkVar (rec_pte_id::context_hyps_ids)
         )
       in
(*      observe_tac "rec hyp " *)
         (tclTHENS
         (Proofview.V82.of_tactic (assert_before (Name rec_pte_id) t_x))
         [
    (* observe_tac "prove rec hyp" *) (prove_rec_hyp eq_hyps);
(* observe_tac "prove rec hyp" *)
     (refine to_refine)
         ])
         g
    )
  ]
     in
     tclTHENLIST
       [
(* observe_tac "hyp rec"  *)
    (change_hyp_with_using "rec_hyp_tac" hyp_id real_type_of_hyp prove_new_type_of_hyp);
  scan_type context popped_t'
       ]
   end
else if eq_constr sigma t_x coq_False then
   begin
(*     observe (str "Removing : "++ Ppconstr.pr_id hyp_id++  *)
(*        str " since it has False in its preconds " *)
(*     ); *)
     raise TOREMOVE;  (* False -> .. useless *)
   end
        else if is_incompatible_eq env sigma t_x then raise TOREMOVE (* t_x := C1 ... =  C2 ... *)
else if eq_constr sigma t_x coq_True  (* Trivial => we remove this precons *)
then
(*     observe (str "In "++Ppconstr.pr_id hyp_id++  *)
(*        str " removing useless precond True" *)
(*     );  *)
   let popped_t' = pop t' in
   let real_type_of_hyp =
     it_mkProd_or_LetIn popped_t' context
   in
   let prove_trivial =
     let nb_intro = List.length context in
     tclTHENLIST [
       tclDO nb_intro (Proofview.V82.of_tactic intro);
       (fun g ->
   let context_hyps =
     fst (list_chop ~msg:"removing True : context_hyps "nb_intro (pf_ids_of_hyps g))
   in
   let to_refine =
     applist (mkVar hyp_id,
       List.rev (coq_I::List.map mkVar context_hyps)
      )
   in
   refine to_refine g
       )
     ]
   in
   tclTHENLIST[
     change_hyp_with_using "prove_trivial" hyp_id real_type_of_hyp
       ((* observe_tac "prove_trivial" *) prove_trivial);
     scan_type context popped_t'
   ]
else if is_trivial_eq sigma t_x
then (*  t_x :=  t = t   => we remove this precond *)
   let popped_t' = pop t' in
   let real_type_of_hyp =
     it_mkProd_or_LetIn popped_t' context
   in
   let hd,args = destApp sigma t_x in
   let get_args hd args =
     if eq_constr sigma hd (Lazy.force eq)
     then (Lazy.force refl_equal,args.(0),args.(1))
     else (jmeq_refl (),args.(0),args.(1))
   in
   tclTHENLIST
     [
       change_hyp_with_using
  "prove_trivial_eq"
  hyp_id
  real_type_of_hyp
  ((* observe_tac "prove_trivial_eq" *)
    (prove_trivial_eq hyp_id context (get_args hd args)));
       scan_type context popped_t'
     ]
else
   begin
     try
       let new_context,new_t',tac = change_eq env sigma hyp_id context x t_x t' in
       tclTHEN
  tac
  (scan_type new_context new_t')
     with NoChange ->
       (* Last thing todo : push the rel in the context and continue *)
              scan_type (LocalAssum (x,t_x) :: context) t'
   end
      end
    else
      tclIDTAC
  in
  try
    scan_type [] (Typing.unsafe_type_of env sigma (mkVar hyp_id)), [hyp_id]
  with TOREMOVE ->
    thin [hyp_id],[]

let clean_goal_with_heq ptes_infos continue_tac (dyn_infos:body_info) =
  fun g ->
    let env = pf_env g
    and sigma = project g
    in
    let tac,new_hyps =
      List.fold_left (
fun (hyps_tac,new_hyps) hyp_id ->
   let hyp_tac,new_hyp =
     clean_hyp_with_heq ptes_infos dyn_infos.eq_hyps hyp_id env sigma
   in
   (tclTHEN hyp_tac hyps_tac),new_hyp@new_hyps
      )
(tclIDTAC,[])
dyn_infos.rec_hyps
    in
    let new_infos =
      { dyn_infos with
   rec_hyps = new_hyps;
   nb_rec_hyps  = List.length new_hyps
      }
    in
    tclTHENLIST
      [
tac ;
(* observe_tac "clean_hyp_with_heq continue" *) (continue_tac new_infos)
      ]
      g

let heq_id = Id.of_string "Heq"

let treat_new_case ptes_infos nb_prod continue_tac term dyn_infos =
  fun g ->
    let nb_first_intro = nb_prod - 1 - dyn_infos.nb_rec_hyps in
    tclTHENLIST
      [
(* We first introduce the variables *)
tclDO nb_first_intro (Proofview.V82.of_tactic (intro_avoiding (Id.Set.of_list dyn_infos.rec_hyps)));
(* Then the equation itself *)
Proofview.V82.of_tactic (intro_using heq_id);
onLastHypId (fun heq_id -> tclTHENLIST [
(* Then the new hypothesis *)
        tclMAP (fun id -> Proofview.V82.of_tactic (introduction id)) dyn_infos.rec_hyps;
observe_tac "after_introduction" (fun g' ->
    (* We get infos on the equations introduced*)
    let new_term_value_eq = pf_unsafe_type_of g' (mkVar heq_id) in
    (* compute the new value of the body *)
    let new_term_value =
      match EConstr.kind (project g') new_term_value_eq with
        | App(f,[| _;_;args2 |]) -> args2
        | _ ->
     observe (str "cannot compute new term value : " ++ pr_gls g' ++ fnl () ++ str "last hyp is" ++
         pr_leconstr_env (pf_env g') (project g') new_term_value_eq
      );
     anomaly (Pp.str "cannot compute new term value.")
    in
  let fun_body =
           mkLambda(make_annot Anonymous Sorts.Relevant,
      pf_unsafe_type_of g' term,
      Termops.replace_term (project g') term (mkRel 1) dyn_infos.info
     )
  in
  let new_body = pf_nf_betaiota g' (mkApp(fun_body,[| new_term_value |])) in
  let new_infos =
    {dyn_infos with
       info = new_body;
       eq_hyps = heq_id::dyn_infos.eq_hyps
    }
  in
  clean_goal_with_heq ptes_infos continue_tac new_infos  g'
      )])
    ]
      g

let my_orelse tac1 tac2 g =
  try
    tac1 g
  with e when CErrors.noncritical e ->
(*     observe (str "using snd tac since : " ++ CErrors.print e); *)
    tac2 g

let instantiate_hyps_with_args (do_prove:Id.t list -> tactic) hyps args_id =
  let args = Array.of_list (List.map mkVar  args_id) in
  let instantiate_one_hyp hid =
    my_orelse
      ( (* we instantiate the hyp if possible  *)
fun g ->
   let prov_hid = pf_get_new_id hid g in
   let c = mkApp(mkVar hid,args) in
   let evm, _ = pf_apply Typing.type_of g c in
   tclTHENLIST[
            Refiner.tclEVARS evm;
     Proofview.V82.of_tactic (pose_proof (Name prov_hid) c);
     thin [hid];
     Proofview.V82.of_tactic (rename_hyp [prov_hid,hid])
   ] g
      )
      ( (*
  if not then we are in a mutual function block
  and this hyp is a recursive hyp on an other function.

  We are not supposed to use it while proving this
  principle so that we can trash it

*)
(fun g ->
(*    observe (str "Instantiation: removing hyp " ++ Ppconstr.pr_id hid); *)
    thin [hid] g
)
      )
  in
  if List.is_empty args_id
  then
    tclTHENLIST [
      tclMAP (fun hyp_id -> h_reduce_with_zeta (Locusops.onHyp hyp_id)) hyps;
      do_prove hyps
    ]
  else
    tclTHENLIST
      [
tclMAP (fun hyp_id -> h_reduce_with_zeta (Locusops.onHyp hyp_id)) hyps;
        tclMAP instantiate_one_hyp hyps;
(fun g ->
    let all_g_hyps_id =
      List.fold_right Id.Set.add (pf_ids_of_hyps g) Id.Set.empty
    in
    let remaining_hyps =
      List.filter (fun id -> Id.Set.mem id all_g_hyps_id) hyps
    in
    do_prove remaining_hyps g
   )
      ]

let build_proof
    (interactive_proof:bool)
    (fnames:Constant.t list)
    ptes_infos
    dyn_infos
    : tactic =
  let rec build_proof_aux do_finalize dyn_infos : tactic =
    fun g ->
        let env = pf_env g in
        let sigma = project g in
(*      observe (str "proving on " ++ Printer.pr_lconstr_env (pf_env g) term);*)
match EConstr.kind sigma dyn_infos.info with
   | Case(ci,ct,t,cb) ->
       let do_finalize_t dyn_info' =
  fun g ->
    let t = dyn_info'.info in
    let dyn_infos = {dyn_info' with info =
        mkCase(ci,ct,t,cb)} in
    let g_nb_prod = nb_prod (project g) (pf_concl g) in
    let type_of_term = pf_unsafe_type_of g t in
    let term_eq =
      make_refl_eq (Lazy.force refl_equal) type_of_term t
    in
    tclTHENLIST
      [
        Proofview.V82.of_tactic (generalize (term_eq::(List.map mkVar dyn_infos.rec_hyps)));
        thin dyn_infos.rec_hyps;
        Proofview.V82.of_tactic (pattern_option [Locus.AllOccurrencesBut [1],t] None);
        (fun g -> observe_tac "toto" (
    tclTHENLIST [Proofview.V82.of_tactic (Simple.case t);
         (fun g' ->
     let g'_nb_prod = nb_prod (project g') (pf_concl g') in
                                        let nb_instantiate_partial = g'_nb_prod - g_nb_prod in
      observe_tac "treat_new_case"
       (treat_new_case
          ptes_infos
                                             nb_instantiate_partial
                                             (build_proof do_finalize)
          t
          dyn_infos)
       g'
         )

        ]) g
        )
      ]
      g
       in
              build_proof do_finalize_t {dyn_infos with info = t} g
          | Lambda(n,t,b) ->
       begin
  match EConstr.kind sigma (pf_concl g) with
    | Prod _ ->
        tclTHEN
   (Proofview.V82.of_tactic intro)
   (fun g' ->
                           let open Context.Named.Declaration in
      let id = pf_last_hyp g' |> get_id in
      let new_term =
        pf_nf_betaiota g'
          (mkApp(dyn_infos.info,[|mkVar id|]))
      in
      let new_infos = {dyn_infos with info = new_term} in
      let do_prove new_hyps =
                             build_proof do_finalize
          {new_infos with
             rec_hyps = new_hyps;
      nb_rec_hyps  = List.length new_hyps
          }
      in
(*    observe_tac "Lambda" *) (instantiate_hyps_with_args do_prove new_infos.rec_hyps [id]) g'
        (*    build_proof do_finalize new_infos g' *)
   ) g
    | _ ->
        do_finalize dyn_infos g
       end
   | Cast(t,_,_) ->
              build_proof do_finalize {dyn_infos with info = t} g
          | Const _ | Var _ | Meta _ | Evar _ | Sort _ | Construct _ | Ind _ | Int _ ->
       do_finalize dyn_infos g
   | App(_,_) ->
       let f,args = decompose_app sigma dyn_infos.info in
       begin
  match EConstr.kind sigma f with
      | Int _ -> user_err Pp.(str "integer cannot be applied")
    | App _ -> assert false (* we have collected all the app in decompose_app *)
    | Proj _ -> assert false (*FIXME*)
    | Var _ | Construct _ | Rel _ | Evar _ | Meta _  | Ind _ | Sort _ | Prod _ ->
        let new_infos =
   { dyn_infos with
       info = (f,args)
   }
        in
                      build_proof_args env sigma do_finalize new_infos  g
    | Const (c,_) when not (List.mem_f Constant.equal c fnames) ->
        let new_infos =
   { dyn_infos with
       info = (f,args)
   }
        in
(*       Pp.msgnl (str "proving in " ++ pr_lconstr_env (pf_env g) dyn_infos.info); *)
                      build_proof_args env sigma do_finalize new_infos g
    | Const _ ->
        do_finalize dyn_infos  g
    | Lambda _ ->
        let new_term =
                        Reductionops.nf_beta env sigma dyn_infos.info in
                      build_proof do_finalize {dyn_infos with info = new_term}
   g
    | LetIn _ ->
        let new_infos =
                        { dyn_infos with info = Reductionops.nf_betaiotazeta env sigma dyn_infos.info }
        in

        tclTHENLIST
   [tclMAP
      (fun hyp_id ->
        h_reduce_with_zeta (Locusops.onHyp hyp_id))
      dyn_infos.rec_hyps;
    h_reduce_with_zeta Locusops.onConcl;
                         build_proof do_finalize new_infos
   ]
   g
    | Cast(b,_,_) ->
                      build_proof do_finalize {dyn_infos with info = b } g
    | Case _ | Fix _ | CoFix _ ->
        let new_finalize dyn_infos =
   let new_infos =
     { dyn_infos with
         info = dyn_infos.info,args
     }
   in
                        build_proof_args env sigma do_finalize new_infos
        in
                      build_proof new_finalize {dyn_infos  with info = f } g
       end
   | Fix _ | CoFix _ ->
       user_err Pp.(str ( "Anonymous local (co)fixpoints are not handled yet"))

   | Proj _ -> user_err Pp.(str "Prod")
   | Prod _ -> do_finalize dyn_infos g
   | LetIn _ ->
       let new_infos =
  { dyn_infos with
                    info = Reductionops.nf_betaiotazeta env sigma dyn_infos.info
  }
       in

       tclTHENLIST
  [tclMAP
     (fun hyp_id -> h_reduce_with_zeta (Locusops.onHyp hyp_id))
     dyn_infos.rec_hyps;
   h_reduce_with_zeta Locusops.onConcl;
                 build_proof do_finalize new_infos
  ] g
   | Rel _ -> anomaly (Pp.str "Free var in goal conclusion!")
  and build_proof do_finalize dyn_infos g =
(*     observe (str "proving with "++Printer.pr_lconstr dyn_infos.info++ str " on goal " ++ pr_gls g); *)
    observe_tac_stream (str "build_proof with " ++ pr_leconstr_env (pf_env g) (project g) dyn_infos.info ) (build_proof_aux do_finalize dyn_infos) g
  and build_proof_args env sigma do_finalize dyn_infos (* f_args'  args *) :tactic =
    fun g ->
      let (f_args',args) = dyn_infos.info in
      let tac : tactic =
fun g ->
   match args with
     | []  ->
       do_finalize {dyn_infos with info = f_args'} g
     | arg::args ->
       (* observe (str "build_proof_args with arg := "++ pr_lconstr_env (pf_env g) arg++ *)
       (* fnl () ++  *)
       (* pr_goal (Tacmach.sig_it g) *)
       (* ); *)
       let do_finalize dyn_infos =
  let new_arg = dyn_infos.info in
  (* tclTRYD *)
                (build_proof_args env sigma
     do_finalize
     {dyn_infos with info = (mkApp(f_args',[|new_arg|])), args}
  )
       in
              build_proof do_finalize
  {dyn_infos with info = arg }
  g
      in
      (* observe_tac "build_proof_args" *) (tac ) g
  in
  let do_finish_proof dyn_infos =
     (* tclTRYD *) (clean_goal_with_heq
        ptes_infos
        finish_proof dyn_infos)
  in
    (* observe_tac "build_proof" *)
  fun g ->
    build_proof (clean_goal_with_heq ptes_infos do_finish_proof) dyn_infos g

(* Proof of principles from structural functions *)

type static_fix_info =
    {
      idx : int;
      name : Id.t;
      types : types;
      offset : int;
      nb_realargs : int;
      body_with_param : constr;
      num_in_block : int
    }

let prove_rec_hyp_for_struct fix_info =
      (fun  eq_hyps -> tclTHEN
(rewrite_until_var (fix_info.idx) eq_hyps)
(fun g ->
    let _,pte_args = destApp (project g) (pf_concl g) in
    let rec_hyp_proof =
      mkApp(mkVar fix_info.name,array_get_start pte_args)
    in
    refine rec_hyp_proof g
))

let prove_rec_hyp fix_info  =
  { proving_tac = prove_rec_hyp_for_struct fix_info
  ;
    is_valid = fun _ -> true
  }

let generalize_non_dep hyp g =
(*   observe (str "rec id := " ++ Ppconstr.pr_id hyp); *)
  let hyps = [hyp] in
  let env = Global.env () in
  let hyp_typ = pf_unsafe_type_of g (mkVar hyp) in
  let to_revert,_ =
    let open Context.Named.Declaration in
    Environ.fold_named_context_reverse (fun (clear,keep) decl ->
      let decl = map_named_decl EConstr.of_constr decl in
      let hyp = get_id decl in
      if Id.List.mem hyp hyps
        || List.exists (Termops.occur_var_in_decl env (project g) hyp) keep
|| Termops.occur_var env (project g) hyp hyp_typ
|| Termops.is_section_variable hyp (* should be dangerous *)
      then (clear,decl::keep)
      else (hyp::clear,keep))
      ~init:([],[]) (pf_env g)
  in
(*   observe (str "to_revert := " ++ prlist_with_sep spc Ppconstr.pr_id to_revert); *)
  tclTHEN
    ((* observe_tac "h_generalize" *) (Proofview.V82.of_tactic (generalize  (List.map mkVar to_revert) )))
    ((* observe_tac "thin" *) (thin to_revert))
    g

let id_of_decl = RelDecl.get_name %> Nameops.Name.get_id
let var_of_decl = id_of_decl %> mkVar
let revert idl =
  tclTHEN
    (Proofview.V82.of_tactic (generalize (List.map mkVar idl)))
    (thin idl)

let generate_equation_lemma evd fnames f fun_num nb_params nb_args rec_args_num =
(*   observe (str "nb_args := " ++ str (string_of_int nb_args)); *)
(*   observe (str "nb_params := " ++ str (string_of_int nb_params)); *)
(*   observe (str "rec_args_num := " ++ str (string_of_int (rec_args_num + 1) )); *)
  let f_def = Global.lookup_constant (fst (destConst evd f)) in
  let eq_lhs = mkApp(f,Array.init (nb_params + nb_args) (fun i -> mkRel(nb_params + nb_args - i))) in
  let (f_body, _) = Option.get (Global.body_of_constant_body f_def) in
  let f_body = EConstr.of_constr f_body in
  let params,f_body_with_params = decompose_lam_n evd nb_params f_body in
  let (_,num),(_,_,bodies) = destFix evd f_body_with_params in
  let fnames_with_params =
    let params = Array.init nb_params (fun i -> mkRel(nb_params - i)) in
    let fnames = List.rev (Array.to_list (Array.map (fun f -> mkApp(f,params)) fnames)) in
    fnames
  in
(*   observe (str "fnames_with_params " ++ prlist_with_sep fnl pr_lconstr fnames_with_params); *)
(*   observe (str "body " ++ pr_lconstr bodies.(num)); *)
  let f_body_with_params_and_other_fun  = substl fnames_with_params bodies.(num) in
(*   observe (str "f_body_with_params_and_other_fun " ++  pr_lconstr f_body_with_params_and_other_fun); *)
  let eq_rhs = Reductionops.nf_betaiotazeta (Global.env ()) evd (mkApp(compose_lam params f_body_with_params_and_other_fun,Array.init (nb_params + nb_args) (fun i -> mkRel(nb_params + nb_args - i)))) in
  (*   observe (str "eq_rhs " ++  pr_lconstr eq_rhs); *)
  let (type_ctxt,type_of_f),evd =
    let evd,t = Typing.type_of ~refresh:true (Global.env ()) evd f
    in
    decompose_prod_n_assum evd
      (nb_params + nb_args) t,evd
  in
  let eqn = mkApp(Lazy.force eq,[|type_of_f;eq_lhs;eq_rhs|]) in
  let lemma_type = it_mkProd_or_LetIn eqn type_ctxt in
  (* Pp.msgnl (str "lemma type " ++ Printer.pr_lconstr lemma_type ++ fnl () ++ str "f_body " ++ Printer.pr_lconstr f_body); *)
  let f_id = Label.to_id (Constant.label (fst (destConst evd f))) in
  let prove_replacement =
    tclTHENLIST
      [
tclDO (nb_params + rec_args_num + 1) (Proofview.V82.of_tactic intro);
observe_tac "" (fun g ->
    let rec_id = pf_nth_hyp_id g 1 in
    tclTHENLIST
      [observe_tac "generalize_non_dep in generate_equation_lemma" (generalize_non_dep rec_id);
       observe_tac "h_case" (Proofview.V82.of_tactic (simplest_case (mkVar rec_id)));
       (Proofview.V82.of_tactic intros_reflexivity)] g
)
      ]
  in
  (* Pp.msgnl (str "lemma type (2) " ++ Printer.pr_lconstr_env (Global.env ()) evd lemma_type); *)
  let pstate = Lemmas.start_proof ~ontop:None
    (*i The next call to mk_equation_id is valid since we are constructing the lemma
      Ensures by: obvious
      i*)
    (mk_equation_id f_id)
    (Decl_kinds.Global, false, (Decl_kinds.Proof Decl_kinds.Theorem))
    evd
  lemma_type
  in
  let pstate,_ = Pfedit.by (Proofview.V82.tactic prove_replacement) pstate in
  let pstate = Lemmas.save_proof_proved ?proof:None ~pstate ~opaque:Proof_global.Transparent ~idopt:None in
  pstate, evd

let do_replace (evd:Evd.evar_map ref) params rec_arg_num rev_args_id f fun_num all_funs g =
  let equation_lemma =
    try
      let finfos = find_Function_infos (fst (destConst !evd f)) (*FIXME*) in
      mkConst (Option.get finfos.equation_lemma)
    with (Not_found | Option.IsNone as e) ->
      let f_id = Label.to_id (Constant.label (fst (destConst !evd f))) in
      (*i The next call to mk_equation_id is valid since we will construct the lemma
Ensures by: obvious
i*)
      let equation_lemma_id = (mk_equation_id f_id) in
      evd := snd @@ generate_equation_lemma !evd all_funs  f fun_num (List.length params) (List.length rev_args_id) rec_arg_num;
      let _ =
match e with
   | Option.IsNone ->
       let finfos = find_Function_infos (fst (destConst !evd f)) in
       update_Function
  {finfos with
     equation_lemma = Some (match Nametab.locate (qualid_of_ident equation_lemma_id) with
           ConstRef c -> c
         | _ -> CErrors.anomaly (Pp.str "Not a constant.")
      )
  }
   | _ -> ()
      in
      (* let res = Constrintern.construct_reference (pf_hyps g) equation_lemma_id in *)
      let evd',res =
Evd.fresh_global
   (Global.env ()) !evd
   (Constrintern.locate_reference (qualid_of_ident equation_lemma_id))
      in
      evd:=evd';
      let sigma, _ = Typing.type_of ~refresh:true (Global.env ()) !evd res in
      evd := sigma;
      res
  in
  let nb_intro_to_do = nb_prod (project g) (pf_concl g) in
    tclTHEN
      (tclDO nb_intro_to_do (Proofview.V82.of_tactic intro))
      (
fun g' ->
   let just_introduced = nLastDecls nb_intro_to_do g' in
          let open Context.Named.Declaration in
   let just_introduced_id = List.map get_id just_introduced in
   tclTHEN (Proofview.V82.of_tactic (Equality.rewriteLR equation_lemma))
    (revert just_introduced_id) g'
      )
      g

let prove_princ_for_struct (evd:Evd.evar_map ref) interactive_proof fun_num fnames all_funs _nparams : tactic =
  fun g ->
  let princ_type = pf_concl g in
  (* Pp.msgnl (str "princ_type " ++ Printer.pr_lconstr princ_type); *)
  (* Pp.msgnl (str "all_funs "); *)
  (* Array.iter (fun c -> Pp.msgnl (Printer.pr_lconstr c)) all_funs; *)
    let princ_info = compute_elim_sig (project g) princ_type in
    let fresh_id =
      let avoid = ref (pf_ids_of_hyps g) in
      (fun na ->
  let new_id =
    match na with
        Name id -> fresh_id !avoid (Id.to_string id)
      | Anonymous -> fresh_id !avoid "H"
  in
  avoid := new_id :: !avoid;
  (Name new_id)
      )
    in
    let fresh_decl = RelDecl.map_name fresh_id in
    let princ_info : elim_scheme =
      { princ_info with
   params = List.map fresh_decl princ_info.params;
   predicates = List.map fresh_decl princ_info.predicates;
   branches = List.map fresh_decl princ_info.branches;
   args = List.map fresh_decl princ_info.args
      }
    in
    let get_body const =
      match Global.body_of_constant const with
| Some (body, _) ->
          let env = Global.env () in
          let sigma = Evd.from_env env in
      Tacred.cbv_norm_flags
        (CClosure.RedFlags.mkflags [CClosure.RedFlags.fZETA])
               env
               sigma
        (EConstr.of_constr body)
| None -> user_err Pp.(str "Cannot define a principle over an axiom ")
    in
    let fbody = get_body fnames.(fun_num) in
    let f_ctxt,f_body = decompose_lam (project g) fbody in
    let f_ctxt_length = List.length f_ctxt in
    let diff_params = princ_info.nparams - f_ctxt_length in
    let full_params,princ_params,fbody_with_full_params =
      if diff_params > 0
      then
let princ_params,full_params =
   list_chop  diff_params princ_info.params
in
(full_params, (* real params *)
  princ_params, (* the params of the principle which are not params of the function *)
         substl (* function instantiated with real params *)
    (List.map var_of_decl full_params)
    f_body
)
      else
let f_ctxt_other,f_ctxt_params =
   list_chop (- diff_params) f_ctxt in
let f_body = compose_lam f_ctxt_other f_body in
(princ_info.params, (* real params *)
  [],(* all params are full params *)
         substl (* function instantiated with real params *)
    (List.map var_of_decl princ_info.params)
    f_body
)
    in
    observe (str "full_params := " ++
        prlist_with_sep spc (RelDecl.get_name %> Nameops.Name.get_id %> Ppconstr.pr_id)
        full_params
     );
    observe (str "princ_params := " ++
        prlist_with_sep spc (RelDecl.get_name %> Nameops.Name.get_id %> Ppconstr.pr_id)
        princ_params
     );
    observe (str "fbody_with_full_params := " ++
               pr_leconstr_env (Global.env ()) !evd fbody_with_full_params
     );
    let all_funs_with_full_params =
      Array.map (fun f -> applist(f, List.rev_map var_of_decl full_params)) all_funs
    in
    let fix_offset = List.length princ_params in
    let ptes_to_fix,infos =
      match EConstr.kind (project g) fbody_with_full_params with
        | Fix((idxs,i),(names,typess,bodies)) ->
     let bodies_with_all_params =
       Array.map
  (fun body ->
                   Reductionops.nf_betaiota (pf_env g) (project g)
       (applist(substl (List.rev (Array.to_list all_funs_with_full_params)) body,
         List.rev_map var_of_decl princ_params))
  )
  bodies
     in
     let info_array =
       Array.mapi
  (fun i types ->
     let types = prod_applist (project g) types (List.rev_map var_of_decl princ_params) in
     { idx = idxs.(i)  - fix_offset;
                     name = Nameops.Name.get_id (fresh_id names.(i).binder_name);
       types = types;
       offset = fix_offset;
       nb_realargs =
         List.length
    (fst (decompose_lam (project g) bodies.(i))) - fix_offset;
       body_with_param = bodies_with_all_params.(i);
       num_in_block = i
     }
  )
  typess
     in
     let pte_to_fix,rev_info =
       List.fold_left_i
  (fun i (acc_map,acc_info) decl ->
     let pte = RelDecl.get_name decl in
     let infos = info_array.(i) in
     let type_args,_ = decompose_prod (project g) infos.types in
     let nargs = List.length type_args in
     let f = applist(mkConst fnames.(i), List.rev_map var_of_decl princ_info.params) in
     let first_args = Array.init nargs (fun i -> mkRel (nargs -i)) in
     let app_f = mkApp(f,first_args) in
     let pte_args = (Array.to_list first_args)@[app_f] in
     let app_pte = applist(mkVar (Nameops.Name.get_id pte),pte_args) in
     let body_with_param,num =
       let body = get_body fnames.(i) in
       let body_with_full_params =
                       Reductionops.nf_betaiota (pf_env g) (project g) (
    applist(body,List.rev_map var_of_decl full_params))
       in
       match EConstr.kind (project g) body_with_full_params with
                       | Fix((_,num),(_,_,bs)) ->
                               Reductionops.nf_betaiota (pf_env g) (project g)
                                 (
       (applist
          (substl
      (List.rev
         (Array.to_list all_funs_with_full_params))
      bs.(num),
           List.rev_map var_of_decl princ_params))
     ),num
    | _ -> user_err Pp.(str "Not a mutual block")
     in
     let info =
       {infos with
   types = compose_prod type_args app_pte;
    body_with_param = body_with_param;
    num_in_block = num
       }
     in
(*    observe (str "binding " ++ Ppconstr.pr_id (Nameops.Name.get_id pte) ++  *)
(*       str " to " ++ Ppconstr.pr_id info.name); *)
     (Id.Map.add (Nameops.Name.get_id pte) info acc_map,info::acc_info)
     )
  0
  (Id.Map.empty,[])
  (List.rev princ_info.predicates)
     in
     pte_to_fix,List.rev rev_info
| _ ->
    Id.Map.empty,[]
    in
    let mk_fixes : tactic =
      let pre_info,infos = list_chop fun_num infos in
      match pre_info,infos with
| _,[] -> tclIDTAC
| _, this_fix_info::others_infos ->
     let other_fix_infos =
       List.map
  (fun fi -> fi.name,fi.idx + 1 ,fi.types)
  (pre_info@others_infos)
     in
     if List.is_empty other_fix_infos
     then
       if this_fix_info.idx + 1 = 0
       then tclIDTAC (* Someone  tries to defined a principle on a fully parametric definition declared as a fixpoint (strange but ....) *)
       else
                observe_tac_stream (str "h_fix " ++ int (this_fix_info.idx +1) ) (Proofview.V82.of_tactic (fix this_fix_info.name (this_fix_info.idx +1)))
     else
       Proofview.V82.of_tactic (Tactics.mutual_fix this_fix_info.name (this_fix_info.idx + 1)
  other_fix_infos 0)
    in
    let first_tac : tactic = (* every operations until fix creations *)
      tclTHENLIST
[ observe_tac "introducing params" (Proofview.V82.of_tactic (intros_using (List.rev_map id_of_decl princ_info.params)));
   observe_tac "introducing predictes" (Proofview.V82.of_tactic (intros_using (List.rev_map id_of_decl princ_info.predicates)));
   observe_tac "introducing branches" (Proofview.V82.of_tactic (intros_using (List.rev_map id_of_decl princ_info.branches)));
   observe_tac "building fixes" mk_fixes;
]
    in
    let intros_after_fixes : tactic =
      fun gl ->
let ctxt,pte_app =  (decompose_prod_assum (project gl) (pf_concl gl)) in
let pte,pte_args = (decompose_app (project gl) pte_app) in
try
   let pte =
            try destVar (project gl) pte
            with DestKO -> anomaly (Pp.str "Property is not a variable.")
          in
   let fix_info = Id.Map.find  pte ptes_to_fix in
   let nb_args = fix_info.nb_realargs in
   tclTHENLIST
     [
       (* observe_tac ("introducing args") *) (tclDO nb_args (Proofview.V82.of_tactic intro));
       (fun g -> (* replacement of the function by its body *)
   let args = nLastDecls nb_args g in
   let fix_body = fix_info.body_with_param in
(* observe (str "fix_body := "++ pr_lconstr_env (pf_env gl) fix_body); *)
                 let open Context.Named.Declaration in
   let args_id = List.map get_id args in
   let dyn_infos =
     {
       nb_rec_hyps = -100;
       rec_hyps = [];
       info =
                       Reductionops.nf_betaiota (pf_env g) (project g)
    (applist(fix_body,List.rev_map mkVar args_id));
       eq_hyps = []
     }
   in
   tclTHENLIST
     [
       observe_tac "do_replace"
         (do_replace evd
     full_params
     (fix_info.idx + List.length princ_params)
     (args_id@(List.map (RelDecl.get_name %> Nameops.Name.get_id) princ_params))
     (all_funs.(fix_info.num_in_block))
     fix_info.num_in_block
     all_funs
         );
       let do_prove =
         build_proof
    interactive_proof
    (Array.to_list fnames)
    (Id.Map.map prove_rec_hyp ptes_to_fix)
       in
       let prove_tac branches  =
         let dyn_infos =
    {dyn_infos with
       rec_hyps = branches;
       nb_rec_hyps = List.length branches
    }
         in
         observe_tac "cleaning" (clean_goal_with_heq
    (Id.Map.map prove_rec_hyp ptes_to_fix)
    do_prove
    dyn_infos)
       in
(*      observe (str "branches := " ++ *)
(* prlist_with_sep spc (fun decl -> Ppconstr.pr_id (id_of_decl decl)) princ_info.branches ++  fnl () ++ *)
(*    str "args := " ++ prlist_with_sep spc Ppconstr.pr_id  args_id *)

(*    ); *)
                     (* observe_tac "instancing" *) (instantiate_hyps_with_args prove_tac
         (List.rev_map id_of_decl princ_info.branches)
         (List.rev args_id))
     ]
     g
       );
     ] gl
with Not_found ->
   let nb_args = min (princ_info.nargs) (List.length ctxt) in
   tclTHENLIST
     [
       tclDO nb_args (Proofview.V82.of_tactic intro);
       (fun g -> (* replacement of the function by its body *)
          let args = nLastDecls nb_args g in
                 let open Context.Named.Declaration in
   let args_id = List.map get_id args in
   let dyn_infos =
     {
       nb_rec_hyps = -100;
       rec_hyps = [];
       info =
                       Reductionops.nf_betaiota (pf_env g) (project g)
    (applist(fbody_with_full_params,
      (List.rev_map var_of_decl princ_params)@
        (List.rev_map mkVar args_id)
     ));
       eq_hyps = []
     }
   in
   let fname = destConst (project g) (fst (decompose_app (project g) (List.hd (List.rev pte_args)))) in
   tclTHENLIST
     [Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, Names.EvalConstRef (fst fname))]);
      let do_prove =
        build_proof
   interactive_proof
   (Array.to_list fnames)
    (Id.Map.map prove_rec_hyp ptes_to_fix)
      in
      let prove_tac branches  =
        let dyn_infos =
    {dyn_infos with
       rec_hyps = branches;
       nb_rec_hyps = List.length branches
    }
        in
         clean_goal_with_heq
    (Id.Map.map prove_rec_hyp ptes_to_fix)
    do_prove
    dyn_infos
      in
                    instantiate_hyps_with_args prove_tac
         (List.rev_map id_of_decl princ_info.branches)
        (List.rev args_id)
     ]
     g
       )
     ]
   gl
    in
    tclTHEN
      first_tac
      intros_after_fixes
      g

(* Proof of principles of general functions *)
(* let  hrec_id = Recdef.hrec_id *)
(* and acc_inv_id = Recdef.acc_inv_id *)
(* and ltof_ref = Recdef.ltof_ref *)
(* and acc_rel = Recdef.acc_rel *)
(* and well_founded = Recdef.well_founded *)
(* and list_rewrite = Recdef.list_rewrite *)
(* and evaluable_of_global_reference = Recdef.evaluable_of_global_reference *)

let prove_with_tcc tcc_lemma_constr eqs : tactic =
  match !tcc_lemma_constr with
  | Undefined -> anomaly (Pp.str "No tcc proof !!")
  | Value lemma ->
fun gls ->
(*   let hid = next_ident_away_in_goal h_id (pf_ids_of_hyps gls) in *)
(*   let ids = hid::pf_ids_of_hyps gls in  *)
   tclTHENLIST
     [
(*       generalize [lemma]; *)
(*       h_intro hid; *)
(*       Elim.h_decompose_and (mkVar hid); *)
       tclTRY(list_rewrite true eqs);
(*       (fun g ->  *)
(* let ids' = pf_ids_of_hyps g in  *)
(* let ids = List.filter (fun id -> not (List.mem id ids)) ids' in  *)
(* rewrite *)
(*       ) *)
       Proofview.V82.of_tactic (Eauto.gen_eauto (false,5) [] (Some []))
     ]
     gls
  | Not_needed -> tclIDTAC

let backtrack_eqs_until_hrec hrec eqs : tactic =
  fun gls ->
    let eqs = List.map mkVar eqs in
    let rewrite =
      tclFIRST (List.map (fun x -> Proofview.V82.of_tactic (Equality.rewriteRL x)) eqs )
    in
    let _,hrec_concl  = decompose_prod (project gls) (pf_unsafe_type_of gls (mkVar hrec)) in
    let f_app = Array.last (snd (destApp (project gls) hrec_concl)) in
    let f =  (fst (destApp (project gls) f_app)) in
    let rec backtrack : tactic =
      fun g ->
let f_app = Array.last (snd (destApp (project g) (pf_concl g))) in
match EConstr.kind (project g) f_app with
   | App(f',_) when eq_constr (project g) f' f -> tclIDTAC g
   | _ -> tclTHEN rewrite backtrack g
    in
    backtrack gls

let rec rewrite_eqs_in_eqs eqs =
  match eqs with
    | [] -> tclIDTAC
    | eq::eqs ->

   tclTHEN
     (tclMAP
        (fun id gl ->
    observe_tac
      (Format.sprintf "rewrite %s in %s " (Id.to_string eq) (Id.to_string id))
      (tclTRY (Proofview.V82.of_tactic (Equality.general_rewrite_in true Locus.AllOccurrences
          true (* dep proofs also: *) true id (mkVar eq) false)))
      gl
        )
        eqs
     )
     (rewrite_eqs_in_eqs eqs)

let new_prove_with_tcc is_mes acc_inv hrec tcc_hyps eqs : tactic =
  fun gls ->
    (tclTHENLIST
       [
  backtrack_eqs_until_hrec hrec eqs;
  (* observe_tac ("new_prove_with_tcc ( applying "^(Id.to_string hrec)^" )" ) *)
  (tclTHENS  (* We must have exactly ONE subgoal !*)
     (Proofview.V82.of_tactic (apply (mkVar hrec)))
     [ tclTHENLIST
  [
    (Proofview.V82.of_tactic (keep (tcc_hyps@eqs)));
    (Proofview.V82.of_tactic (apply (Lazy.force acc_inv)));
    (fun g ->
       if is_mes
       then
         Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, evaluable_of_global_reference (delayed_force ltof_ref))]) g
       else tclIDTAC g
    );
    observe_tac "rew_and_finish"
      (tclTHENLIST
         [tclTRY(list_rewrite false (List.map (fun v -> (mkVar v,true)) eqs));
   observe_tac "rewrite_eqs_in_eqs" (rewrite_eqs_in_eqs eqs);
    (observe_tac "finishing using"
      (
       tclCOMPLETE(
        Eauto.eauto_with_bases
          (true,5)
          [(fun _ sigma -> (sigma, Lazy.force refl_equal))]
                                      [Hints.Hint_db.empty TransparentState.empty false]
      )
      )
   )
         ]
      )
  ]
     ])
       ])
      gls

let is_valid_hypothesis sigma predicates_name =
  let predicates_name = List.fold_right Id.Set.add predicates_name Id.Set.empty in
  let is_pte typ =
    if isApp sigma typ
    then
      let pte,_ = destApp sigma typ in
      if isVar sigma pte
      then Id.Set.mem (destVar sigma pte) predicates_name
      else false
    else false
  in
  let rec is_valid_hypothesis typ =
    is_pte typ ||
      match EConstr.kind sigma typ with
        | Prod(_,pte,typ') -> is_pte pte && is_valid_hypothesis typ'
| _ -> false
  in
  is_valid_hypothesis

let prove_principle_for_gen
    (f_ref,functional_ref,eq_ref) tcc_lemma_ref is_mes
    rec_arg_num rec_arg_type relation gl =
  let princ_type = pf_concl gl in
  let princ_info = compute_elim_sig (project gl) princ_type in
  let fresh_id =
    let avoid = ref (pf_ids_of_hyps gl) in
    fun na ->
      let new_id =
   match na with
     | Name id -> fresh_id !avoid (Id.to_string id)
     | Anonymous -> fresh_id !avoid "H"
      in
      avoid := new_id :: !avoid;
      Name new_id
  in
  let fresh_decl = map_name fresh_id in
  let princ_info : elim_scheme =
    { princ_info with
params = List.map fresh_decl princ_info.params;
predicates = List.map fresh_decl princ_info.predicates;
branches = List.map fresh_decl princ_info.branches;
args = List.map fresh_decl princ_info.args
    }
  in
  let wf_tac =
    if is_mes
    then
      (fun b -> Recdef.tclUSER_if_not_mes tclIDTAC b None)
    else fun _ -> prove_with_tcc tcc_lemma_ref []
  in
  let real_rec_arg_num = rec_arg_num - princ_info.nparams in
  let npost_rec_arg = princ_info.nargs - real_rec_arg_num + 1 in
(*   observe ( *)
(*     str "princ_type := " ++ pr_lconstr  princ_type ++ fnl () ++ *)
(*     str "princ_info.nparams := " ++ int princ_info.nparams ++ fnl () ++  *)

(*       str "princ_info.nargs := " ++ int princ_info.nargs ++ fnl () ++  *)
(*     str "rec_arg_num := " ++ int rec_arg_num ++ fnl() ++  *)
(*      str "real_rec_arg_num := " ++ int real_rec_arg_num ++ fnl () ++ *)
(*      str "npost_rec_arg := " ++ int npost_rec_arg ); *)
  let (post_rec_arg,pre_rec_arg) =
    Util.List.chop npost_rec_arg princ_info.args
  in
  let rec_arg_id =
    match List.rev post_rec_arg with
      | (LocalAssum ({binder_name=Name id},_) | LocalDef ({binder_name=Name id},_,_)) :: _ -> id
      | _ -> assert false
  in
(*   observe (str "rec_arg_id := " ++ pr_lconstr (mkVar rec_arg_id)); *)
  let subst_constrs = List.map (get_name %> Nameops.Name.get_id %> mkVar) (pre_rec_arg@princ_info.params) in
  let relation = substl subst_constrs relation in
  let input_type = substl subst_constrs rec_arg_type in
  let wf_thm_id = Nameops.Name.get_id (fresh_id (Name (Id.of_string "wf_R"))) in
  let acc_rec_arg_id =
    Nameops.Name.get_id (fresh_id (Name (Id.of_string ("Acc_"^(Id.to_string rec_arg_id)))))
  in
  let revert l =
    tclTHEN (Proofview.V82.of_tactic (Tactics.generalize (List.map mkVar l))) (Proofview.V82.of_tactic (clear l))
  in
  let fix_id = Nameops.Name.get_id (fresh_id (Name hrec_id)) in
  let prove_rec_arg_acc g =
      ((* observe_tac "prove_rec_arg_acc"  *)
  (tclCOMPLETE
     (tclTHEN
        (Proofview.V82.of_tactic (assert_by (Name wf_thm_id)
    (mkApp (delayed_force well_founded,[|input_type;relation|]))
    (Proofview.V82.tactic (fun g -> (* observe_tac "prove wf" *) (tclCOMPLETE (wf_tac is_mes)) g))))
        (
   (* observe_tac  *)
(*    "apply wf_thm"  *)
   Proofview.V82.of_tactic (Tactics.Simple.apply (mkApp(mkVar wf_thm_id,[|mkVar rec_arg_id|])))
        )
     )
  )
      )
      g
  in
  let args_ids = List.map (get_name %> Nameops.Name.get_id) princ_info.args in
  let lemma =
    match !tcc_lemma_ref with
     | Undefined -> user_err Pp.(str "No tcc proof !!")
     | Value lemma -> EConstr.of_constr lemma
     | Not_needed -> EConstr.of_constr (UnivGen.constr_of_monomorphic_global @@ Coqlib.lib_ref "core.True.I")
  in
(*   let rec list_diff del_list check_list = *)
(*     match del_list with *)
(*          [] -> *)
(*            [] *)
(*       | f::r -> *)
(*           if List.mem f check_list then *)
(*                list_diff r check_list *)
(*           else *)
(*                f::(list_diff r check_list) *)
(*   in *)
  let tcc_list = ref [] in
  let start_tac gls =
    let hyps = pf_ids_of_hyps gls in
      let hid =
next_ident_away_in_goal
   (Id.of_string "prov")
   (Id.Set.of_list hyps)
      in
      tclTHENLIST
[
   Proofview.V82.of_tactic (generalize [lemma]);
   Proofview.V82.of_tactic (Simple.intro hid);
   Proofview.V82.of_tactic (Elim.h_decompose_and (mkVar hid));
   (fun g ->
      let new_hyps = pf_ids_of_hyps g in
      tcc_list := List.rev (List.subtract Id.equal new_hyps (hid::hyps));
      if List.is_empty !tcc_list
      then
        begin
   tcc_list := [hid];
   tclIDTAC g
        end
      else thin [hid] g
   )
   ]
   gls
  in
  tclTHENLIST
    [
      observe_tac "start_tac" start_tac;
      h_intros
(List.rev_map (get_name %> Nameops.Name.get_id)
    (princ_info.args@princ_info.branches@princ_info.predicates@princ_info.params)
);
      (* observe_tac "" *) Proofview.V82.of_tactic (assert_by
  (Name acc_rec_arg_id)
   (mkApp (delayed_force acc_rel,[|input_type;relation;mkVar rec_arg_id|]))
  (Proofview.V82.tactic prove_rec_arg_acc)
      );
(*       observe_tac "reverting" *) (revert (List.rev (acc_rec_arg_id::args_ids)));
(*       (fun g -> observe (Printer.pr_goal (sig_it g) ++ fnl () ++  *)
(*    str "fix arg num" ++ int (List.length args_ids + 1) ); tclIDTAC g); *)
      (* observe_tac "h_fix " *) (Proofview.V82.of_tactic (fix fix_id (List.length args_ids + 1)));
(*       (fun g -> observe (Printer.pr_goal (sig_it g) ++ fnl() ++ pr_lconstr_env (pf_env g ) (pf_unsafe_type_of g (mkVar fix_id) )); tclIDTAC g); *)
      h_intros (List.rev (acc_rec_arg_id::args_ids));
      Proofview.V82.of_tactic (Equality.rewriteLR (mkConst eq_ref));
      (* observe_tac "finish" *) (fun gl' ->
  let body =
    let _,args = destApp (project gl') (pf_concl gl') in
    Array.last args
  in
  let body_info rec_hyps =
    {
      nb_rec_hyps = List.length rec_hyps;
      rec_hyps = rec_hyps;
      eq_hyps = [];
      info = body
    }
  in
  let acc_inv =
    lazy (
      mkApp (
        delayed_force acc_inv_id,
        [|input_type;relation;mkVar rec_arg_id|]
      )
    )
  in
  let acc_inv = lazy (mkApp(Lazy.force acc_inv, [|mkVar  acc_rec_arg_id|])) in
  let predicates_names =
    List.map (get_name %> Nameops.Name.get_id) princ_info.predicates
  in
  let pte_info =
    { proving_tac =
        (fun eqs ->
(*   msgnl (str "tcc_list := "++ prlist_with_sep spc Ppconstr.pr_id  !tcc_list); *)
(*   msgnl (str "princ_info.args := "++ prlist_with_sep spc Ppconstr.pr_id  (List.map  (fun (na,_,_) -> (Nameops.Name.get_id na)) princ_info.args)); *)
(*   msgnl (str "princ_info.params := "++ prlist_with_sep spc Ppconstr.pr_id  (List.map  (fun (na,_,_) -> (Nameops.Name.get_id na)) princ_info.params)); *)
(*   msgnl (str "acc_rec_arg_id := "++  Ppconstr.pr_id acc_rec_arg_id); *)
(*   msgnl (str "eqs := "++ prlist_with_sep spc Ppconstr.pr_id  eqs); *)

    (* observe_tac "new_prove_with_tcc"  *)
      (new_prove_with_tcc
         is_mes acc_inv fix_id

         (!tcc_list@(List.map
      (get_name %> Nameops.Name.get_id)
      (princ_info.args@princ_info.params)
   )@ ([acc_rec_arg_id])) eqs
      )

        );
      is_valid = is_valid_hypothesis (project gl') predicates_names
    }
  in
  let ptes_info : pte_info Id.Map.t =
    List.fold_left
      (fun map pte_id ->
  Id.Map.add pte_id
    pte_info
    map
      )
      Id.Map.empty
      predicates_names
  in
  let make_proof rec_hyps =
    build_proof
      false
      [f_ref]
      ptes_info
      (body_info rec_hyps)
  in
         (* observe_tac "instantiate_hyps_with_args"  *)
           (instantiate_hyps_with_args
       make_proof
       (List.map (get_name %> Nameops.Name.get_id) princ_info.branches)
       (List.rev args_ids)
    )
    gl'
      )

    ]
    gl

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