Quellcode-Bibliothek

^© Kompilation durch diese Firma

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Datei: law_cos_pos_2D.pvs Sprache: SML

Original von: Coq^©

(************************************************************************)
(*         *   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)         *)
(************************************************************************)

module CVars = Vars

open Constr
open Context
open EConstr
open Vars
open Namegen
open Environ
open Entries
open Pp
open Names
open Libnames
open Globnames
open Nameops
open CErrors
open Util
open UnivGen
open Tacticals
open Tacmach
open Tactics
open Nametab
open Declare
open Decl_kinds
open Tacred
open Goal
open Pfedit
open Glob_term
open Pretyping
open Termops
open Constrintern
open Tactypes
open Genredexpr

open Equality
open Auto
open Eauto

open Indfun_common
open Context.Rel.Declaration

(* Ugly things which should not be here *)

[@@@ocaml.warning "-3"]
let coq_constant m s = EConstr.of_constr @@ UnivGen.constr_of_monomorphic_global @@
  Coqlib.find_reference "RecursiveDefinition" m s

let arith_Nat = ["Coq"; "Arith";"PeanoNat";"Nat"]
let arith_Lt  = ["Coq"; "Arith";"Lt"]

let coq_init_constant s =
  EConstr.of_constr (
    UnivGen.constr_of_monomorphic_global @@
    Coqlib.gen_reference_in_modules "RecursiveDefinition" Coqlib.init_modules s)
[@@@ocaml.warning "+3"]

let find_reference sl s =
  let dp = Names.DirPath.make (List.rev_map Id.of_string sl) in
  locate (make_qualid dp (Id.of_string s))

let declare_fun f_id kind ?univs value =
  let ce = definition_entry ?univs value (*FIXME *) in
    ConstRef(declare_constant f_id (DefinitionEntry ce, kind));;

let defined pstate = Lemmas.save_proof_proved ?proof:None ~pstate ~opaque:Proof_global.Transparent ~idopt:None

let def_of_const t =
   match (Constr.kind t) with
    Const sp ->
      (try (match constant_opt_value_in (Global.env ()) sp with
             | Some c -> c
      | _ -> raise Not_found)
       with Not_found ->
  anomaly (str "Cannot find definition of constant " ++
      (Id.print (Label.to_id (Constant.label (fst sp)))) ++ str ".")
      )
     |_ -> assert false

let type_of_const sigma t =
   match (EConstr.kind sigma t) with
   | Const (sp, u) ->
      let u = EInstance.kind sigma u in
      (* FIXME discarding universe constraints *)
      Typeops.type_of_constant_in (Global.env()) (sp, u)
    |_ -> assert false

let constant sl s = UnivGen.constr_of_monomorphic_global (find_reference sl s)

let const_of_ref = function
    ConstRef kn -> kn
  | _ -> anomaly (Pp.str "ConstRef expected.")

(* Generic values *)
let pf_get_new_ids idl g =
  let ids = pf_ids_of_hyps g in
  let ids = Id.Set.of_list ids in
  List.fold_right
    (fun id acc -> next_global_ident_away id (Id.Set.union (Id.Set.of_list acc) ids)::acc)
    idl
    []

let next_ident_away_in_goal ids avoid =
  next_ident_away_in_goal ids (Id.Set.of_list avoid)

let compute_renamed_type gls c =
  rename_bound_vars_as_displayed (project gls) (*no avoid*) Id.Set.empty (*no rels*) []
    (pf_unsafe_type_of gls c)
let h'_id = Id.of_string "h'"
let teq_id = Id.of_string "teq"
let ano_id = Id.of_string "anonymous"
let x_id = Id.of_string "x"
let k_id = Id.of_string "k"
let v_id = Id.of_string "v"
let def_id = Id.of_string "def"
let p_id = Id.of_string "p"
let rec_res_id = Id.of_string "rec_res";;
let lt = function () -> (coq_init_constant "lt")
[@@@ocaml.warning "-3"]
let le = function () -> (Coqlib.gen_reference_in_modules "RecursiveDefinition" Coqlib.init_modules "le")
let ex = function () -> (coq_init_constant "ex")
let nat = function () -> (coq_init_constant "nat")
let iter_ref () =
  try find_reference ["Recdef"] "iter"
  with Not_found -> user_err Pp.(str "module Recdef not loaded")
let iter_rd = function () -> (constr_of_global (delayed_force iter_ref))
let eq = function () -> (coq_init_constant "eq")
let le_lt_SS = function () -> (constant ["Recdef"] "le_lt_SS")
let le_lt_n_Sm = function () -> (coq_constant arith_Lt "le_lt_n_Sm")
let le_trans = function () -> (coq_constant arith_Nat "le_trans")
let le_lt_trans = function () -> (coq_constant arith_Nat "le_lt_trans")
let lt_S_n = function () -> (coq_constant arith_Lt "lt_S_n")
let le_n = function () -> (coq_init_constant "le_n")
let coq_sig_ref = function () -> (find_reference ["Coq";"Init";"Specif"] "sig")
let coq_O = function () -> (coq_init_constant "O")
let coq_S = function () -> (coq_init_constant "S")
let lt_n_O = function () -> (coq_constant arith_Nat "nlt_0_r")
let max_ref = function () -> (find_reference ["Recdef"] "max")
let max_constr = function () -> EConstr.of_constr (constr_of_global (delayed_force max_ref))

let f_S t = mkApp(delayed_force coq_S, [|t|]);;

let rec n_x_id ids n =
  if Int.equal n 0 then []
  else let x = next_ident_away_in_goal x_id ids in
          x::n_x_id (x::ids) (n-1);;

let simpl_iter clause =
  reduce
    (Lazy
       {rBeta=true;rMatch=true;rFix=true;rCofix=true;rZeta=true;rDelta=false;
        rConst = [ EvalConstRef (const_of_ref (delayed_force iter_ref))]})
    clause

(* Others ugly things ... *)
let (value_f: Constr.t list -> GlobRef.t -> Constr.t) =
  let open Term in
  let open Constr in
  fun al fterm ->
    let rev_x_id_l =
      (
List.fold_left
   (fun x_id_l _ ->
      let x_id = next_ident_away_in_goal x_id x_id_l in
      x_id::x_id_l
   )
   []
   al
      )
    in
    let context = List.map
      (fun (x, c) -> LocalAssum (make_annot (Name x) Sorts.Relevant, c)) (List.combine rev_x_id_l (List.rev al))
    in
    let env = Environ.push_rel_context context (Global.env ()) in
    let glob_body =
      DAst.make @@
       GCases
(RegularStyle,None,
  [DAst.make @@ GApp(DAst.make @@ GRef(fterm,None), List.rev_map (fun x_id -> DAst.make @@ GVar x_id) rev_x_id_l),
   (Anonymous,None)],
         [CAst.make ([v_id], [DAst.make @@ PatCstr ((destIndRef (delayed_force coq_sig_ref),1),
      [DAst.make @@ PatVar(Name v_id); DAst.make @@ PatVar Anonymous],
                           Anonymous)],
     DAst.make @@ GVar v_id)])
    in
    let body = fst (understand env (Evd.from_env env) glob_body)(*FIXME*) in
    let body = EConstr.Unsafe.to_constr body in
    it_mkLambda_or_LetIn body context

let (declare_f : Id.t -> logical_kind -> Constr.t list -> GlobRef.t -> GlobRef.t) =
  fun f_id kind input_type fterm_ref ->
    declare_fun f_id kind (value_f input_type fterm_ref);;

(* Debugging mechanism *)
let debug_queue = Stack.create ()

let print_debug_queue b e =
  if  not (Stack.is_empty debug_queue)
  then
    begin
      let lmsg,goal = Stack.pop debug_queue in
      if b then
Feedback.msg_debug (hov 1 (lmsg ++ (str " raised exception " ++ CErrors.print e) ++ str " on goal" ++ fnl() ++ goal))
      else
begin
   Feedback.msg_debug (hov 1 (str " from " ++ lmsg ++ str " on goal"++fnl() ++ goal));
end;
      (* print_debug_queue false e; *)
    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 s = s (pf_env g) (project g) in
  let lmsg = (str "recdef : ") ++ s in
  observe (s++fnl());
  Stack.push (lmsg,goal) debug_queue;
  try
    let v = tac g in
    ignore(Stack.pop debug_queue);
    v
  with reraise ->
    let reraise = CErrors.push reraise in
    if not (Stack.is_empty debug_queue)
    then print_debug_queue true (fst (ExplainErr.process_vernac_interp_error reraise));
    iraise reraise

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

let observe_tclTHENLIST s tacl =
  if do_observe ()
  then
    let rec aux n = function
      | [] -> tclIDTAC
      | [tac] -> observe_tac (fun env sigma -> s env sigma ++ spc () ++ int n) tac
      | tac::tacl -> observe_tac (fun env sigma -> s env sigma ++ spc () ++ int n) (tclTHEN tac (aux (succ n) tacl))
    in
    aux 0 tacl
  else tclTHENLIST tacl

(* Conclusion tactics *)

(* The boolean value is_mes expresses that the termination is expressed
  using a measure function instead of a well-founded relation. *)
let tclUSER tac is_mes l g =
  let clear_tac =
    match l with
      | None -> tclIDTAC
      | Some l -> tclMAP (fun id -> tclTRY (Proofview.V82.of_tactic (clear [id]))) (List.rev l)
  in
  observe_tclTHENLIST (fun _ _ -> str "tclUSER1")
    [
      clear_tac;
      if is_mes
      then observe_tclTHENLIST (fun _ _ -> str "tclUSER2")
        [
   Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, evaluable_of_global_reference
            (delayed_force Indfun_common.ltof_ref))]);
         tac
  ]
      else tac
    ]
    g

let tclUSER_if_not_mes concl_tac is_mes names_to_suppress =
  if is_mes
  then tclCOMPLETE (fun gl -> Proofview.V82.of_tactic (Simple.apply (delayed_force well_founded_ltof)) gl)
  else (* tclTHEN (Simple.apply (delayed_force acc_intro_generator_function) ) *) (tclUSER concl_tac is_mes names_to_suppress)



(* Traveling term.
   Both definitions of [f_terminate] and [f_equation] use the same generic
   traveling mechanism.
*)

(* [check_not_nested forbidden e] checks that [e] does not contains any variable
   of [forbidden]
*)
let check_not_nested env sigma forbidden e =
  let rec check_not_nested e =
    match EConstr.kind sigma e with
      | Rel _ -> ()
      | Int _ -> ()
      | Var x ->
        if Id.List.mem x forbidden
        then user_err ~hdr:"Recdef.check_not_nested"
               (str "check_not_nested: failure " ++ Id.print x)
      | Meta _ | Evar _ | Sort _ -> ()
      | Cast(e,_,t) -> check_not_nested e;check_not_nested t
      | Prod(_,t,b) -> check_not_nested t;check_not_nested b
      | Lambda(_,t,b) -> check_not_nested t;check_not_nested b
      | LetIn(_,v,t,b) -> check_not_nested t;check_not_nested b;check_not_nested v
      | App(f,l) -> check_not_nested f;Array.iter check_not_nested l
      | Proj (p,c) -> check_not_nested c
      | Const _ -> ()
      | Ind _ -> ()
      | Construct _ -> ()
      | Case(_,t,e,a) ->
check_not_nested t;check_not_nested e;Array.iter check_not_nested a
      | Fix _ -> user_err Pp.(str "check_not_nested : Fix")
      | CoFix _ -> user_err Pp.(str "check_not_nested : Fix")
  in
  try
    check_not_nested e
  with UserError(_,p) ->
    user_err ~hdr:"_" (str "on expr : " ++ Printer.pr_leconstr_env env sigma e ++ str " " ++ p)

(* ['a info] contains the local information for traveling *)
type 'a infos =
    { nb_arg : int; (* function number of arguments *)
      concl_tac : tactic; (* final tactic to finish proofs *)
      rec_arg_id : Id.t; (*name of the declared recursive argument *)
      is_mes : bool; (* type of recursion *)
      ih : Id.t; (* induction hypothesis name *)
      f_id : Id.t;  (* function name *)
      f_constr : constr;  (* function term *)
      f_terminate : constr; (* termination proof term *)
      func : GlobRef.t; (* functional reference *)
      info : 'a;
      is_main_branch : bool; (* on the main branch or on a matched expression *)
      is_final : bool; (* final first order term or not *)
      values_and_bounds : (Id.t*Id.t) list;
      eqs : Id.t list;
      forbidden_ids : Id.t list;
      acc_inv : constr lazy_t;
      acc_id : Id.t;
      args_assoc : ((constr list)*constr) list;
    }

type ('a,'b) journey_info_tac =
    'a -> (* the arguments of the constructor *)
    'b infos -> (* infos of the caller *)
    ('b infos -> tactic) -> (* the continuation tactic of the caller *)
    'b infos -> (* argument of the tactic *)
    tactic

(* journey_info : specifies the actions to do on the different term constructors during the traveling of the term
*)
type journey_info =
    { letiN : ((Name.t*constr*types*constr),constr) journey_info_tac;
      lambdA : ((Name.t*types*constr),constr) journey_info_tac;
      casE : ((constr infos -> tactic) -> constr infos -> tactic) ->
((case_info * constr * constr * constr array),constr) journey_info_tac;
      otherS : (unit,constr) journey_info_tac;
      apP : (constr*(constr list),constr) journey_info_tac;
      app_reC : (constr*(constr list),constr) journey_info_tac;
      message : string
    }



let add_vars sigma forbidden e =
  let rec aux forbidden e =
  match EConstr.kind sigma e with
    | Var x -> x::forbidden
    | _ -> EConstr.fold sigma aux forbidden e
  in
  aux forbidden e

let treat_case forbid_new_ids to_intros finalize_tac nb_lam e infos : tactic =
  fun g ->
    let rev_context,b = decompose_lam_n (project g) nb_lam e in
    let ids = List.fold_left (fun acc (na,_) ->
      let pre_id =
        match na.binder_name with
   | Name x -> x
   | Anonymous -> ano_id
      in
      pre_id::acc
    ) [] rev_context in
    let rev_ids = pf_get_new_ids (List.rev ids) g in
    let new_b = substl (List.map mkVar rev_ids) b in
    observe_tclTHENLIST (fun _ _ -> str "treat_case1")
      [
h_intros (List.rev rev_ids);
Proofview.V82.of_tactic (intro_using teq_id);
onLastHypId (fun heq ->
          observe_tclTHENLIST (fun _ _ -> str "treat_case2")[
     Proofview.V82.of_tactic (clear to_intros);
     h_intros to_intros;
     (fun g' ->
       let ty_teq = pf_unsafe_type_of g' (mkVar heq) in
       let teq_lhs,teq_rhs =
  let _,args = try destApp (project g') ty_teq with DestKO -> assert false in
  args.(1),args.(2)
       in
       let new_b' = Termops.replace_term (project g') teq_lhs teq_rhs new_b in
       let new_infos = {
  infos with
    info = new_b';
    eqs = heq::infos.eqs;
    forbidden_ids =
    if forbid_new_ids
    then add_vars (project g') infos.forbidden_ids new_b'
    else infos.forbidden_ids
       } in
       finalize_tac new_infos g'
     )
   ]
)
      ] g

let rec travel_aux jinfo continuation_tac (expr_info:constr infos) g =
  let sigma = project g in
  let env = pf_env g in
  match EConstr.kind sigma expr_info.info with
    | CoFix _ | Fix _ -> user_err Pp.(str "Function cannot treat local fixpoint or cofixpoint")
    | Proj _ -> user_err Pp.(str "Function cannot treat projections")
    | LetIn(na,b,t,e) ->
      begin
let new_continuation_tac =
          jinfo.letiN (na.binder_name,b,t,e) expr_info continuation_tac
in
travel jinfo new_continuation_tac
   {expr_info with info = b; is_final=false} g
      end
    | Rel _ -> anomaly (Pp.str "Free var in goal conclusion!")
    | Prod _ ->
      begin
try
          check_not_nested env sigma (expr_info.f_id::expr_info.forbidden_ids) expr_info.info;
   jinfo.otherS () expr_info continuation_tac expr_info g
with e when CErrors.noncritical e ->
          user_err ~hdr:"Recdef.travel" (str "the term " ++ Printer.pr_leconstr_env env sigma expr_info.info ++ str " can not contain a recursive call to " ++ Id.print expr_info.f_id)
      end
    | Lambda(n,t,b) ->
      begin
try
          check_not_nested env sigma (expr_info.f_id::expr_info.forbidden_ids) expr_info.info;
   jinfo.otherS () expr_info continuation_tac expr_info g
with e when CErrors.noncritical e ->
          user_err ~hdr:"Recdef.travel" (str "the term " ++ Printer.pr_leconstr_env env sigma expr_info.info ++ str " can not contain a recursive call to " ++ Id.print expr_info.f_id)
      end
    | Case(ci,t,a,l) ->
      begin
let continuation_tac_a =
   jinfo.casE
     (travel jinfo) (ci,t,a,l)
     expr_info continuation_tac in
travel
   jinfo continuation_tac_a
   {expr_info with info = a; is_main_branch = false;
     is_final = false} g
      end
    | App _ ->
      let f,args = decompose_app sigma expr_info.info in
      if EConstr.eq_constr sigma f (expr_info.f_constr)
      then jinfo.app_reC (f,args) expr_info continuation_tac expr_info g
      else
      begin
match EConstr.kind sigma f with
   | App _ -> assert false (* f is coming from a decompose_app *)
   | Const _ | Construct _ | Rel _ | Evar _ | Meta _  | Ind _
   | Sort _ | Prod _ | Var _ ->
     let new_infos = {expr_info with info=(f,args)} in
     let new_continuation_tac =
       jinfo.apP (f,args) expr_info continuation_tac in
     travel_args jinfo
       expr_info.is_main_branch new_continuation_tac new_infos g
          | Case _ ->  user_err ~hdr:"Recdef.travel" (str "the term " ++ Printer.pr_leconstr_env env sigma expr_info.info ++ str " can not contain an applied match (See Limitation in Section 2.3 of refman)")
          | _ -> anomaly (Pp.str "travel_aux : unexpected "++ Printer.pr_leconstr_env env sigma expr_info.info ++ Pp.str ".")
      end
    | Cast(t,_,_) -> travel jinfo continuation_tac {expr_info with info=t} g
    | Const _ | Var _ | Meta _ | Evar _ | Sort _ | Construct _ | Ind _ | Int _ ->
      let new_continuation_tac =
jinfo.otherS () expr_info continuation_tac in
      new_continuation_tac expr_info g
and travel_args jinfo is_final continuation_tac infos =
  let (f_args',args) = infos.info in
  match args with
    | [] ->
      continuation_tac {infos with info = f_args'; is_final = is_final}
    | arg::args' ->
      let new_continuation_tac new_infos =
let new_arg = new_infos.info in
travel_args jinfo is_final
   continuation_tac
   {new_infos with info = (mkApp(f_args',[|new_arg|]),args')}
      in
      travel jinfo new_continuation_tac
{infos with info=arg;is_final=false}
and travel jinfo continuation_tac expr_info =
  observe_tac
    (fun env sigma -> str jinfo.message ++ Printer.pr_leconstr_env env sigma expr_info.info)
    (travel_aux jinfo continuation_tac expr_info)

(* Termination proof *)

let rec prove_lt hyple g =
  let sigma = project g in
  begin
    try
      let (varx,varz) = match decompose_app sigma (pf_concl g) with
        | _, x::z::_ when isVar sigma x && isVar sigma z -> x, z
        | _ -> assert false
      in
      let h =
List.find (fun id ->
          match decompose_app sigma (pf_unsafe_type_of g (mkVar id)) with
            | _, t::_ -> EConstr.eq_constr sigma t varx
            | _ -> false
) hyple
      in
      let y =
List.hd (List.tl (snd (decompose_app sigma (pf_unsafe_type_of g (mkVar h))))) in
      observe_tclTHENLIST (fun _ _ -> str "prove_lt1")[
Proofview.V82.of_tactic (apply (mkApp(le_lt_trans (),[|varx;y;varz;mkVar h|])));
        observe_tac (fun _ _ -> str "prove_lt") (prove_lt hyple)
      ]
    with Not_found ->
      (
(
          observe_tclTHENLIST (fun _ _ -> str "prove_lt2")[
     Proofview.V82.of_tactic (apply (delayed_force lt_S_n));
            (observe_tac (fun _ _ -> str "assumption: " ++ Printer.pr_goal g) (Proofview.V82.of_tactic assumption))
   ])
      )
  end
    g

let rec destruct_bounds_aux infos (bound,hyple,rechyps) lbounds g =
  match lbounds with
    | [] ->
      let ids = pf_ids_of_hyps g in
      let s_max = mkApp(delayed_force coq_S, [|bound|]) in
      let k = next_ident_away_in_goal k_id ids in
      let ids = k::ids in
      let h' = next_ident_away_in_goal (h'_id) ids in
      let ids = h'::ids in
      let def = next_ident_away_in_goal def_id ids in
      observe_tclTHENLIST (fun _ _ -> str "destruct_bounds_aux1")[
Proofview.V82.of_tactic (split (ImplicitBindings [s_max]));
Proofview.V82.of_tactic (intro_then
   (fun id ->
            Proofview.V82.tactic begin
            observe_tac (fun _ _ -> str "destruct_bounds_aux")
       (tclTHENS (Proofview.V82.of_tactic (simplest_case (mkVar id)))
   [
                   observe_tclTHENLIST (fun _ _ -> str "")[Proofview.V82.of_tactic (intro_using h_id);
          Proofview.V82.of_tactic (simplest_elim(mkApp(delayed_force lt_n_O,[|s_max|])));
          Proofview.V82.of_tactic default_full_auto];
                   observe_tclTHENLIST (fun _ _ -> str "destruct_bounds_aux2")[
                     observe_tac (fun _ _ -> str "clearing k ") (Proofview.V82.of_tactic (clear [id]));
       h_intros [k;h';def];
                     observe_tac (fun _ _ -> str "simple_iter") (Proofview.V82.of_tactic (simpl_iter Locusops.onConcl));
                     observe_tac (fun _ _ -> str "unfold functional")
         (Proofview.V82.of_tactic (unfold_in_concl[(Locus.OnlyOccurrences [1],
      evaluable_of_global_reference infos.func)]));
       (
                       observe_tclTHENLIST (fun _ _ -> str "test")[
    list_rewrite true
      (List.fold_right
         (fun e acc -> (mkVar e,true)::acc)
         infos.eqs
         (List.map (fun e -> (e,true)) rechyps)
      );
        (* list_rewrite true *)
        (*   (List.map (fun e -> (mkVar e,true)) infos.eqs) *)
        (*   ; *)

                         (observe_tac (fun _ _ -> str "finishing")
       (tclORELSE
          (Proofview.V82.of_tactic intros_reflexivity)
                               (observe_tac (fun _ _ -> str "calling prove_lt") (prove_lt hyple))))])
     ]
       ]
       )end))
   ] g
      | (_,v_bound)::l ->
      observe_tclTHENLIST (fun _ _ -> str "destruct_bounds_aux3")[
Proofview.V82.of_tactic (simplest_elim (mkVar v_bound));
Proofview.V82.of_tactic (clear [v_bound]);
tclDO 2 (Proofview.V82.of_tactic intro);
onNthHypId 1
   (fun p_hyp ->
     (onNthHypId 2
        (fun p ->
                 observe_tclTHENLIST (fun _ _ -> str "destruct_bounds_aux4")[
     Proofview.V82.of_tactic (simplest_elim
       (mkApp(delayed_force max_constr, [| bound; mkVar p|])));
     tclDO 3 (Proofview.V82.of_tactic intro);
     onNLastHypsId 3 (fun lids ->
       match lids with
         [hle2;hle1;pmax] ->
    destruct_bounds_aux infos
      ((mkVar pmax),
       hle1::hle2::hyple,(mkVar p_hyp)::rechyps)
      l
         | _ -> assert false) ;
   ]
        )
     )
   )
      ] g

let destruct_bounds infos =
  destruct_bounds_aux infos (delayed_force coq_O,[],[]) infos.values_and_bounds

let terminate_app f_and_args expr_info continuation_tac infos =
    if expr_info.is_final && expr_info.is_main_branch
    then
      observe_tclTHENLIST (fun _ _ -> str "terminate_app1")[
continuation_tac infos;
        observe_tac (fun _ _ -> str "first split")
   (Proofview.V82.of_tactic (split (ImplicitBindings [infos.info])));
        observe_tac (fun _ _ -> str "destruct_bounds (1)") (destruct_bounds infos)
      ]
    else continuation_tac infos

let terminate_others _ expr_info continuation_tac infos =
  if expr_info.is_final && expr_info.is_main_branch
  then
    observe_tclTHENLIST (fun _ _ -> str "terminate_others")[
     continuation_tac infos;
      observe_tac (fun _ _ -> str "first split")
(Proofview.V82.of_tactic (split (ImplicitBindings [infos.info])));
      observe_tac (fun _ _ -> str "destruct_bounds") (destruct_bounds infos)
    ]
  else continuation_tac infos

let terminate_letin (na,b,t,e) expr_info continuation_tac info g =
  let sigma = project g in
  let env = pf_env g in
  let new_e = subst1 info.info e in
  let new_forbidden =
    let forbid =
      try
        check_not_nested env sigma (expr_info.f_id::expr_info.forbidden_ids) b;
true
      with e when CErrors.noncritical e -> false
    in
    if forbid
    then
      match na with
| Anonymous -> info.forbidden_ids
| Name id -> id::info.forbidden_ids
    else info.forbidden_ids
  in
  continuation_tac {info with info = new_e; forbidden_ids = new_forbidden} g

let pf_type c tac gl =
  let evars, ty = Typing.type_of (pf_env gl) (project gl) c in
    tclTHEN (Refiner.tclEVARS evars) (tac ty) gl

let pf_typel l tac =
  let rec aux tys l =
    match l with
    | [] -> tac (List.rev tys)
    | hd :: tl -> pf_type hd (fun ty -> aux (ty::tys) tl)
  in aux [] l

(* This is like the previous one except that it also rewrite on all
  hypotheses except the ones given in the first argument.  All the
  modified hypotheses are generalized in the process and should be
  introduced back later; the result is the pair of the tactic and the
  list of hypotheses that have been generalized and cleared. *)
let mkDestructEq :
  Id.t list -> constr -> goal Evd.sigma -> tactic * Id.t list =
  fun not_on_hyp expr g ->
  let hyps = pf_hyps g in
  let to_revert =
    Util.List.map_filter
      (fun decl ->
        let open Context.Named.Declaration in
        let id = get_id decl in
        if Id.List.mem id not_on_hyp || not (Termops.dependent (project g) expr (get_type decl))
        then None else Some id) hyps in
  let to_revert_constr = List.rev_map mkVar to_revert in
  let type_of_expr = pf_unsafe_type_of g expr in
  let new_hyps = mkApp(Lazy.force refl_equal, [|type_of_expr; expr|])::
           to_revert_constr in
    pf_typel new_hyps (fun _ ->
    observe_tclTHENLIST (fun _ _ -> str "mkDestructEq")
     [Proofview.V82.of_tactic (generalize new_hyps);
      (fun g2 ->
        let changefun patvars env sigma =
          pattern_occs [Locus.AllOccurrencesBut [1], expr] (pf_env g2) sigma (pf_concl g2)
        in
Proofview.V82.of_tactic (change_in_concl None changefun) g2);
      Proofview.V82.of_tactic (simplest_case expr)]), to_revert

let terminate_case next_step (ci,a,t,l) expr_info continuation_tac infos g =
  let sigma = project g in
  let env = pf_env g in
  let f_is_present =
    try
      check_not_nested env sigma (expr_info.f_id::expr_info.forbidden_ids) a;
      false
    with e when CErrors.noncritical e ->
      true
  in
  let a' = infos.info in
  let new_info =
    {infos with
      info = mkCase(ci,t,a',l);
      is_main_branch = expr_info.is_main_branch;
      is_final = expr_info.is_final} in
  let destruct_tac,rev_to_thin_intro =
    mkDestructEq [expr_info.rec_arg_id] a' g in
  let to_thin_intro = List.rev rev_to_thin_intro in
  observe_tac (fun _ _ -> str "treating cases (" ++ int (Array.length l) ++ str")" ++ spc () ++ Printer.pr_leconstr_env (pf_env g) sigma a')
    (try
      (tclTHENS
  destruct_tac
         (List.map_i (fun i e -> observe_tac (fun _ _ -> str "do treat case") (treat_case f_is_present to_thin_intro (next_step continuation_tac) ci.ci_cstr_ndecls.(i) e new_info)) 0 (Array.to_list l)
  ))
    with
      | UserError(Some "Refiner.thensn_tac3",_)
      | UserError(Some "Refiner.tclFAIL_s",_) ->
        (observe_tac (fun _ _ -> str "is computable " ++ Printer.pr_leconstr_env env sigma new_info.info) (next_step continuation_tac {new_info with info = Reductionops.nf_betaiotazeta (pf_env g) sigma new_info.info} )
))
    g

let terminate_app_rec (f,args) expr_info continuation_tac _ g =
  let sigma = project g in
  let env = pf_env g in
  List.iter (check_not_nested env sigma (expr_info.f_id::expr_info.forbidden_ids))
    args;
  begin
    try
      let v = List.assoc_f (List.equal (EConstr.eq_constr sigma)) args expr_info.args_assoc in
      let new_infos = {expr_info with info = v} in
      observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec")[
continuation_tac new_infos;
if expr_info.is_final && expr_info.is_main_branch
then
          observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec1")[
            observe_tac (fun _ _ -> str "first split")
       (Proofview.V82.of_tactic (split (ImplicitBindings [new_infos.info])));
            observe_tac (fun _ _ -> str "destruct_bounds (3)")
       (destruct_bounds new_infos)
   ]
else
   tclIDTAC
      ] g
    with Not_found ->
      observe_tac (fun _ _ -> str "terminate_app_rec not found") (tclTHENS
(Proofview.V82.of_tactic (simplest_elim (mkApp(mkVar expr_info.ih,Array.of_list args))))
[
          observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec2")[
     Proofview.V82.of_tactic (intro_using rec_res_id);
     Proofview.V82.of_tactic intro;
     onNthHypId 1
       (fun v_bound ->
  (onNthHypId 2
     (fun v ->
       let new_infos = { expr_info with
         info = (mkVar v);
         values_and_bounds =
    (v,v_bound)::expr_info.values_and_bounds;
         args_assoc=(args,mkVar v)::expr_info.args_assoc
       } in
                     observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec3")[
         continuation_tac new_infos;
         if expr_info.is_final && expr_info.is_main_branch
         then
                         observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec4")[
                           observe_tac (fun _ _ -> str "first split")
        (Proofview.V82.of_tactic (split (ImplicitBindings [new_infos.info])));
                           observe_tac (fun _ _ -> str "destruct_bounds (2)")
        (destruct_bounds new_infos)
    ]
         else
    tclIDTAC
       ]
     )
  )
       )
   ];
          observe_tac (fun _ _ -> str "proving decreasing") (
     tclTHENS (* proof of args < formal args *)
       (Proofview.V82.of_tactic (apply (Lazy.force expr_info.acc_inv)))
       [
                observe_tac (fun _ _ -> str "assumption") (Proofview.V82.of_tactic assumption);
                observe_tclTHENLIST (fun _ _ -> str "terminate_app_rec5")
    [
      tclTRY(list_rewrite true
        (List.map
    (fun e -> mkVar e,true)
    expr_info.eqs
        )
      );
      tclUSER expr_info.concl_tac true
        (Some (
        expr_info.ih::expr_info.acc_id::
   (fun (x,y) -> y)
   (List.split expr_info.values_and_bounds)
         )
        );
    ]
       ])
]) g
  end

let terminate_info =
  { message = "prove_terminate with term ";
    letiN = terminate_letin;
    lambdA = (fun _ _ _ _ -> assert false);
    casE = terminate_case;
    otherS = terminate_others;
    apP = terminate_app;
    app_reC = terminate_app_rec;
  }

let prove_terminate = travel terminate_info

(* Equation proof *)

let equation_case next_step (ci,a,t,l) expr_info continuation_tac infos =
  observe_tac (fun _ _ -> str "equation case") (terminate_case next_step (ci,a,t,l) expr_info continuation_tac infos)

let rec prove_le g =
  let sigma = project g in
  let x,z =
    let _,args = decompose_app sigma (pf_concl g) in
    (List.hd args,List.hd (List.tl args))
  in
  tclFIRST[
    Proofview.V82.of_tactic assumption;
    Proofview.V82.of_tactic (apply (delayed_force le_n));
    begin
      try
        let matching_fun c = match EConstr.kind sigma c with
        | App (c, [| x0 ; _ |]) ->
          EConstr.isVar sigma x0 &&
          Id.equal (destVar sigma x0) (destVar sigma x) &&
          EConstr.is_global sigma (le ()) c
        | _ -> false
        in
        let (h,t) = List.find (fun (_,t) -> matching_fun t) (pf_hyps_types g) in
        let h = h.binder_name in
let y =
   let _,args = decompose_app sigma t in
   List.hd (List.tl args)
in
        observe_tclTHENLIST (fun _ _ -> str "prove_le")[
   Proofview.V82.of_tactic (apply(mkApp(le_trans (),[|x;y;z;mkVar h|])));
          observe_tac (fun _ _ -> str "prove_le (rec)") (prove_le)
]
      with Not_found -> tclFAIL 0 (mt())
    end;
  ]
    g

let rec make_rewrite_list expr_info max = function
  | [] -> tclIDTAC
  | (_,p,hp)::l ->
    observe_tac (fun _ _ -> str "make_rewrite_list") (tclTHENS
      (observe_tac (fun _ _ -> str "rewrite heq on " ++ Id.print p ) (
(fun g ->
          let sigma = project g in
   let t_eq = compute_renamed_type g (mkVar hp) in
   let k,def =
            let k_na,_,t = destProd sigma t_eq in
            let _,_,t  = destProd sigma t in
            let def_na,_,_ = destProd sigma t in
            Nameops.Name.get_id k_na.binder_name,Nameops.Name.get_id def_na.binder_name
   in
   Proofview.V82.of_tactic (general_rewrite_bindings false Locus.AllOccurrences
     true (* dep proofs also: *) true
     (mkVar hp,
             ExplicitBindings[CAst.make @@ (NamedHyp def, expr_info.f_constr);
                              CAst.make @@ (NamedHyp k, f_S max)]) false) g) )
      )
      [make_rewrite_list expr_info max l;
       observe_tclTHENLIST (fun _ _ -> str "make_rewrite_list")[ (* x < S max proof *)
  Proofview.V82.of_tactic (apply (delayed_force le_lt_n_Sm));
         observe_tac (fun _ _ -> str "prove_le(2)") prove_le
       ]
      ] )

let make_rewrite expr_info l hp max =
  tclTHENFIRST
    (observe_tac (fun _ _ -> str "make_rewrite") (make_rewrite_list expr_info max l))
    (observe_tac (fun _ _ -> str "make_rewrite") (tclTHENS
       (fun g ->
          let sigma = project g in
   let t_eq = compute_renamed_type g (mkVar hp) in
   let k,def =
            let k_na,_,t = destProd sigma t_eq in
            let _,_,t  = destProd sigma t in
            let def_na,_,_ = destProd sigma t in
            Nameops.Name.get_id k_na.binder_name,Nameops.Name.get_id def_na.binder_name
   in
         observe_tac (fun _ _ -> str "general_rewrite_bindings")
    (Proofview.V82.of_tactic (general_rewrite_bindings false Locus.AllOccurrences
     true (* dep proofs also: *) true
     (mkVar hp,
             ExplicitBindings[CAst.make @@ (NamedHyp def, expr_info.f_constr);
                              CAst.make @@ (NamedHyp k, f_S (f_S max))]) false)) g)
       [observe_tac(fun _ _ -> str "make_rewrite finalize") (
  (* tclORELSE( h_reflexivity) *)
         (observe_tclTHENLIST (fun _ _ -> str "make_rewrite")[
    Proofview.V82.of_tactic (simpl_iter Locusops.onConcl);
           observe_tac (fun _ _ -> str "unfold functional")
      (Proofview.V82.of_tactic (unfold_in_concl[(Locus.OnlyOccurrences [1],
          evaluable_of_global_reference expr_info.func)]));

    (list_rewrite true
       (List.map (fun e -> mkVar e,true) expr_info.eqs));
           (observe_tac (fun _ _ -> str "h_reflexivity")
   (Proofview.V82.of_tactic intros_reflexivity)
    )
     ]))
       ;
         observe_tclTHENLIST (fun _ _ -> str "make_rewrite1")[ (* x < S (S max) proof *)
  Proofview.V82.of_tactic (apply (EConstr.of_constr (delayed_force le_lt_SS)));
         observe_tac (fun _ _ -> str "prove_le (3)") prove_le
  ]
       ])
    )

let rec compute_max rew_tac max l =
  match l with
    | [] -> rew_tac max
    | (_,p,_)::l ->
      observe_tclTHENLIST (fun _ _ -> str "compute_max")[
Proofview.V82.of_tactic (simplest_elim
   (mkApp(delayed_force max_constr, [| max; mkVar p|])));
tclDO 3 (Proofview.V82.of_tactic intro);
onNLastHypsId 3 (fun lids ->
   match lids with
     | [hle2;hle1;pmax] -> compute_max rew_tac (mkVar pmax) l
     | _ -> assert false
)]

let rec destruct_hex expr_info acc l =
  match l with
    | [] ->
      begin
match List.rev acc with
   | [] -> tclIDTAC
   | (_,p,hp)::tl  ->
            observe_tac (fun _ _ -> str "compute max ") (compute_max (make_rewrite expr_info tl hp) (mkVar p) tl)
      end
    | (v,hex)::l ->
      observe_tclTHENLIST (fun _ _ -> str "destruct_hex")[
Proofview.V82.of_tactic (simplest_case (mkVar hex));
Proofview.V82.of_tactic (clear [hex]);
tclDO 2 (Proofview.V82.of_tactic intro);
onNthHypId 1 (fun hp ->
   onNthHypId 2 (fun p ->
     observe_tac
              (fun _ _ -> str "destruct_hex after " ++ Id.print hp ++ spc () ++ Id.print p)
       (destruct_hex expr_info ((v,p,hp)::acc) l)
   )
)
      ]

let rec intros_values_eq expr_info acc =
  tclORELSE(
    observe_tclTHENLIST (fun _ _ -> str "intros_values_eq")[
      tclDO 2 (Proofview.V82.of_tactic intro);
      onNthHypId 1 (fun hex ->
(onNthHypId 2 (fun v -> intros_values_eq expr_info ((v,hex)::acc)))
      )
    ])
    (tclCOMPLETE (
      destruct_hex expr_info [] acc
    ))

let equation_others _ expr_info continuation_tac infos =
  if expr_info.is_final && expr_info.is_main_branch
  then
    observe_tac (fun env sigma -> str "equation_others (cont_tac +intros) " ++ Printer.pr_leconstr_env env sigma expr_info.info)
  (tclTHEN
      (continuation_tac infos)
      (observe_tac (fun env sigma -> str "intros_values_eq equation_others "  ++ Printer.pr_leconstr_env env sigma expr_info.info) (intros_values_eq expr_info [])))
  else observe_tac (fun env sigma -> str "equation_others (cont_tac) " ++ Printer.pr_leconstr_env env sigma expr_info.info) (continuation_tac infos)

let equation_app f_and_args expr_info continuation_tac infos =
    if expr_info.is_final && expr_info.is_main_branch
    then ((observe_tac (fun _ _ -> str "intros_values_eq equation_app") (intros_values_eq expr_info [])))
    else continuation_tac infos

let equation_app_rec (f,args) expr_info continuation_tac info g =
  let sigma = project g in
  begin
    try
      let v = List.assoc_f (List.equal (EConstr.eq_constr sigma)) args expr_info.args_assoc in
      let new_infos = {expr_info with info = v} in
      observe_tac (fun _ _ -> str "app_rec found") (continuation_tac new_infos) g
    with Not_found ->
      if expr_info.is_final && expr_info.is_main_branch
      then
        observe_tclTHENLIST (fun _ _ -> str "equation_app_rec")
   [ Proofview.V82.of_tactic (simplest_case (mkApp (expr_info.f_terminate,Array.of_list args)));
     continuation_tac {expr_info with args_assoc = (args,delayed_force coq_O)::expr_info.args_assoc};
            observe_tac (fun _ _ -> str "app_rec intros_values_eq") (intros_values_eq expr_info [])
   ] g
      else
        observe_tclTHENLIST (fun _ _ -> str "equation_app_rec1")[
     Proofview.V82.of_tactic (simplest_case (mkApp (expr_info.f_terminate,Array.of_list args)));
          observe_tac (fun _ _ -> str "app_rec not_found") (continuation_tac {expr_info with args_assoc = (args,delayed_force coq_O)::expr_info.args_assoc})
] g
  end

let equation_info =
  {message = "prove_equation with term ";
   letiN = (fun _ -> assert false);
   lambdA = (fun _ _ _ _ -> assert false);
   casE = equation_case;
   otherS = equation_others;
   apP = equation_app;
   app_reC = equation_app_rec
}

let prove_eq = travel equation_info

(* wrappers *)
(* [compute_terminate_type] computes the type of the Definition f_terminate from the type of f_F
*)
let compute_terminate_type nb_args func =
  let open Term in
  let open Constr in
  let open CVars in
  let _,a_arrow_b,_ = destLambda(def_of_const (constr_of_global func)) in
  let rev_args,b = decompose_prod_n nb_args a_arrow_b in
  let left =
    mkApp(delayed_force iter_rd,
   Array.of_list
     (lift 5 a_arrow_b:: mkRel 3::
        constr_of_global func::mkRel 1::
        List.rev (List.map_i (fun i _ -> mkRel (6+i)) 0 rev_args)
     )
  )
  in
  let right = mkRel 5 in
  let delayed_force c = EConstr.Unsafe.to_constr (delayed_force c) in
  let equality = mkApp(delayed_force eq, [|lift 5 b; left; right|]) in
  let result = (mkProd (make_annot (Name def_id) Sorts.Relevant, lift 4 a_arrow_b, equality)) in
  let cond = mkApp(delayed_force lt, [|(mkRel 2); (mkRel 1)|]) in
  let nb_iter =
    mkApp(delayed_force ex,
   [|delayed_force nat;
     (mkLambda
               (make_annot (Name p_id) Sorts.Relevant,
    delayed_force nat,
                  (mkProd (make_annot (Name k_id) Sorts.Relevant, delayed_force nat,
                           mkArrow cond Sorts.Relevant result))))|])in
  let value = mkApp(constr_of_global (Util.delayed_force coq_sig_ref),
      [|b;
                      (mkLambda (make_annot (Name v_id) Sorts.Relevant, b, nb_iter))|]) in
  compose_prod rev_args value

let termination_proof_header is_mes input_type ids args_id relation
    rec_arg_num rec_arg_id tac wf_tac : tactic =
  begin
    fun g ->
      let nargs = List.length args_id in
      let pre_rec_args =
List.rev_map
   mkVar (fst (List.chop (rec_arg_num - 1) args_id))
      in
      let relation = substl pre_rec_args relation in
      let input_type = substl pre_rec_args input_type in
      let wf_thm = next_ident_away_in_goal (Id.of_string ("wf_R")) ids in
      let wf_rec_arg =
next_ident_away_in_goal
   (Id.of_string ("Acc_"^(Id.to_string rec_arg_id)))
   (wf_thm::ids)
      in
      let hrec = next_ident_away_in_goal hrec_id
(wf_rec_arg::wf_thm::ids) in
      let acc_inv =
   lazy (
     mkApp (
       delayed_force acc_inv_id,
       [|input_type;relation;mkVar rec_arg_id|]
     )
   )
      in
      tclTHEN
(h_intros args_id)
(tclTHENS
    (observe_tac
              (fun _ _ -> str "first assert")
       (Proofview.V82.of_tactic (assert_before
   (Name wf_rec_arg)
   (mkApp (delayed_force acc_rel,
    [|input_type;relation;mkVar rec_arg_id|])
   )
       ))
    )
    [
      (* accesibility proof *)
      tclTHENS
        (observe_tac
                  (fun _ _ -> str "second assert")
    (Proofview.V82.of_tactic (assert_before
       (Name wf_thm)
       (mkApp (delayed_force well_founded,[|input_type;relation|]))
    ))
        )
        [
   (* interactive proof that the relation is well_founded *)
                 observe_tac (fun _ _ -> str "wf_tac") (wf_tac is_mes (Some args_id));
   (* this gives the accessibility argument *)
   observe_tac
                   (fun _ _ -> str "apply wf_thm")
     (Proofview.V82.of_tactic (Simple.apply (mkApp(mkVar wf_thm,[|mkVar rec_arg_id|])))
     )
        ]
      ;
      (* rest of the proof *)
             observe_tclTHENLIST (fun _ _ -> str "rest of proof")
               [observe_tac (fun _ _ -> str "generalize")
    (onNLastHypsId (nargs+1)
       (tclMAP (fun id ->
   tclTHEN (Proofview.V82.of_tactic (Tactics.generalize [mkVar id])) (Proofview.V82.of_tactic (clear [id])))
       ))
        ;
                observe_tac (fun _ _ -> str "fix") (Proofview.V82.of_tactic (fix hrec (nargs+1)));
  h_intros args_id;
  Proofview.V82.of_tactic (Simple.intro wf_rec_arg);
                observe_tac (fun _ _ -> str "tac") (tac wf_rec_arg hrec wf_rec_arg acc_inv)
        ]
    ]
) g
  end

let rec instantiate_lambda sigma t l =
  match l with
  | [] -> t
  | a::l ->
      let (_, _, body) = destLambda sigma t in
      instantiate_lambda sigma (subst1 a body) l

let whole_start (concl_tac:tactic) nb_args is_mes func input_type relation rec_arg_num  : tactic =
  begin
    fun g ->
      let sigma = project g in
      let ids = Termops.ids_of_named_context (pf_hyps g) in
      let func_body = (def_of_const (constr_of_global func)) in
      let func_body = EConstr.of_constr func_body in
      let (f_name, _, body1) = destLambda sigma func_body in
      let f_id =
        match f_name.binder_name with
   | Name f_id -> next_ident_away_in_goal f_id ids
   | Anonymous -> anomaly (Pp.str "Anonymous function.")
      in
      let n_names_types,_ = decompose_lam_n sigma nb_args body1 in
      let n_ids,ids =
List.fold_left
          (fun (n_ids,ids) (n_name,_) ->
             match n_name.binder_name with
        | Name id ->
     let n_id = next_ident_away_in_goal id ids in
     n_id::n_ids,n_id::ids
        | _ -> anomaly (Pp.str "anonymous argument.")
   )
   ([],(f_id::ids))
   n_names_types
      in
      let rec_arg_id = List.nth n_ids (rec_arg_num - 1) in
      let expr = instantiate_lambda sigma func_body (mkVar f_id::(List.map mkVar n_ids)) in
      termination_proof_header
is_mes
input_type
ids
n_ids
relation
rec_arg_num
rec_arg_id
(fun rec_arg_id hrec acc_id acc_inv g ->
   (prove_terminate (fun infos -> tclIDTAC)
      { is_main_branch = true; (* we are on the main branche (i.e. still on a match ... with .... end *)
        is_final = true;      (* and on leaf (more or less) *)
        f_terminate = delayed_force coq_O;
        nb_arg = nb_args;
        concl_tac = concl_tac;
        rec_arg_id = rec_arg_id;
        is_mes = is_mes;
        ih = hrec;
        f_id = f_id;
        f_constr = mkVar f_id;
        func = func;
        info = expr;
        acc_inv = acc_inv;
        acc_id = acc_id;
        values_and_bounds = [];
        eqs = [];
        forbidden_ids = [];
        args_assoc = []
      }
   )
      g
)
(tclUSER_if_not_mes concl_tac)
g
  end

let get_current_subgoals_types pstate =
  let p = Proof_global.give_me_the_proof pstate in
  let sgs,_,_,_,sigma = Proof.proof p in
  sigma, List.map (Goal.V82.abstract_type sigma) sgs

exception EmptySubgoals
let build_and_l sigma l =
  let and_constr =  UnivGen.constr_of_monomorphic_global @@ Coqlib.lib_ref "core.and.type" in
  let conj_constr = Coqlib.build_coq_conj () in
  let mk_and p1 p2 =
    mkApp(EConstr.of_constr and_constr,[|p1;p2|]) in
  let rec is_well_founded t =
    match EConstr.kind sigma t with
      | Prod(_,_,t') -> is_well_founded t'
      | App(_,_) ->
let (f,_) = decompose_app sigma t in
EConstr.eq_constr sigma f (well_founded ())
      | _ ->
false
  in
  let compare t1 t2 =
    let b1,b2= is_well_founded t1,is_well_founded t2 in
    if (b1&&b2) || not (b1 || b2) then 0
    else if b1 && not b2 then 1 else -1
  in
  let l = List.sort compare l in
  let rec f  = function
    | [] -> raise EmptySubgoals
    | [p] -> p,tclIDTAC,1
    | p1::pl ->
let c,tac,nb = f pl in
mk_and p1 c,
tclTHENS
   (Proofview.V82.of_tactic (apply (EConstr.of_constr (constr_of_global conj_constr))))
   [tclIDTAC;
    tac
   ],nb+1
  in f l

let is_rec_res id =
  let rec_res_name = Id.to_string rec_res_id   in
  let id_name = Id.to_string id in
  try
    String.equal (String.sub id_name 0 (String.length rec_res_name)) rec_res_name
  with Invalid_argument _ -> false

let clear_goals sigma =
  let rec clear_goal t =
    match EConstr.kind sigma t with
      | Prod({binder_name=Name id} as na,t',b) ->
   let b' = clear_goal b in
   if noccurn sigma 1 b' && (is_rec_res id)
   then Vars.lift (-1) b'
   else if b' == b then t
          else mkProd(na,t',b')
      | _ -> EConstr.map sigma clear_goal t
  in
  List.map clear_goal

let build_new_goal_type pstate =
  let sigma, sub_gls_types = get_current_subgoals_types pstate in
  (* Pp.msgnl (str "sub_gls_types1 := " ++ Util.prlist_with_sep (fun () -> Pp.fnl () ++ Pp.fnl ()) Printer.pr_lconstr sub_gls_types); *)
  let sub_gls_types = clear_goals sigma sub_gls_types in
  (* Pp.msgnl (str "sub_gls_types2 := " ++ Pp.prlist_with_sep (fun () -> Pp.fnl () ++ Pp.fnl ()) Printer.pr_lconstr sub_gls_types); *)
  let res = build_and_l sigma sub_gls_types in
    sigma, res

let is_opaque_constant c =
  let cb = Global.lookup_constant c in
  match cb.Declarations.const_body with
    | Declarations.OpaqueDef _ -> Proof_global.Opaque
    | Declarations.Undef _ -> Proof_global.Opaque
    | Declarations.Def _ -> Proof_global.Transparent
    | Declarations.Primitive _ -> Proof_global.Opaque

let open_new_goal pstate build_proof sigma using_lemmas ref_ goal_name (gls_type,decompose_and_tac,nb_goal)   =
  (* Pp.msgnl (str "gls_type := " ++ Printer.pr_lconstr gls_type); *)
  let current_proof_name = Proof_global.get_current_proof_name pstate in
  let name = match goal_name with
    | Some s -> s
    | None   ->
try add_suffix current_proof_name "_subproof"
with e when CErrors.noncritical e ->
          anomaly (Pp.str "open_new_goal with an unnamed theorem.")
  in
  let na = next_global_ident_away name Id.Set.empty in
  if Termops.occur_existential sigma gls_type then
    CErrors.user_err Pp.(str "\"abstract\" cannot handle existentials");
  let hook _ _ _ _ =
    let opacity =
      let na_ref = qualid_of_ident na in
      let na_global = Smartlocate.global_with_alias na_ref in
      match na_global with
   ConstRef c -> is_opaque_constant c
| _ -> anomaly ~label:"equation_lemma" (Pp.str "not a constant.")
    in
    let lemma = mkConst (Names.Constant.make1 (Lib.make_kn na)) in
    ref_ := Value (EConstr.Unsafe.to_constr lemma);
    let lid = ref [] in
    let h_num = ref (-1) in
    let env = Global.env () in
    let pstate = build_proof env (Evd.from_env env)
      (  fun gls ->
    let hid = next_ident_away_in_goal h_id (pf_ids_of_hyps gls) in
           observe_tclTHENLIST (fun _ _ -> str "")
      [
        Proofview.V82.of_tactic (generalize [lemma]);
        Proofview.V82.of_tactic (Simple.intro hid);
        (fun g ->
    let ids = pf_ids_of_hyps g in
    tclTHEN
      (Proofview.V82.of_tactic (Elim.h_decompose_and (mkVar hid)))
      (fun g ->
         let ids' = pf_ids_of_hyps g in
         lid := List.rev (List.subtract Id.equal ids' ids);
         if List.is_empty !lid then lid := [hid];
         tclIDTAC g
      )
      g
        );
      ] gls)
      (fun g ->
        let sigma = project g in
  match EConstr.kind sigma (pf_concl g) with
    | App(f,_) when EConstr.eq_constr sigma f (well_founded ()) ->
        Proofview.V82.of_tactic (Auto.h_auto None [] (Some []))  g
    | _ ->
        incr h_num;
               (observe_tac (fun _ _ -> str "finishing using")
    (
      tclCOMPLETE(
        tclFIRST[
   tclTHEN
     (Proofview.V82.of_tactic (eapply_with_bindings (mkVar (List.nth !lid !h_num), NoBindings)))
     (Proofview.V82.of_tactic e_assumption);
        Eauto.eauto_with_bases
   (true,5)
   [(fun _ sigma -> (sigma, (Lazy.force refl_equal)))]
                        [Hints.Hint_db.empty TransparentState.empty false]
        ]
      )
    )
        )
         g)
    in
    let _pstate = Lemmas.save_proof_proved ?proof:None ~pstate ~opaque:opacity ~idopt:None in
    ()
  in
  let pstate = Lemmas.start_proof ~ontop:(Some pstate)
    na
    (Decl_kinds.Global, false (* FIXME *), Decl_kinds.Proof Decl_kinds.Lemma)
    sigma gls_type ~hook:(Lemmas.mk_hook hook) in
  let pstate = if Indfun_common.is_strict_tcc  ()
  then
    fst @@ by (Proofview.V82.tactic (tclIDTAC)) pstate
  else
    fst @@ by (Proofview.V82.tactic begin
fun g ->
   tclTHEN
     (decompose_and_tac)
     (tclORELSE
        (tclFIRST
     (List.map
        (fun c ->
    Proofview.V82.of_tactic (Tacticals.New.tclTHENLIST
      [intros;
                           Simple.apply (fst (interp_constr (Global.env()) Evd.empty c)) (*FIXME*);
       Tacticals.New.tclCOMPLETE Auto.default_auto
      ])
        )
        using_lemmas)
        ) tclIDTAC)
            g end) pstate
  in
  try
    Some (fst @@ by (Proofview.V82.tactic tclIDTAC) pstate) (* raises UserError _ if the proof is complete *)
  with UserError _ ->
    defined pstate

let com_terminate
    tcc_lemma_name
    tcc_lemma_ref
    is_mes
    fonctional_ref
    input_type
    relation
    rec_arg_num
    thm_name using_lemmas
    nb_args ctx
    hook =
  let start_proof env ctx (tac_start:tactic) (tac_end:tactic) =
    let pstate = Lemmas.start_proof ~ontop:None thm_name
      (Global, false (* FIXME *), Proof Lemma) ~sign:(Environ.named_context_val env)
      ctx (EConstr.of_constr (compute_terminate_type nb_args fonctional_ref)) ~hook in
    let pstate = fst @@ by (Proofview.V82.tactic (observe_tac (fun _ _ -> str "starting_tac") tac_start)) pstate in
    fst @@ by (Proofview.V82.tactic (observe_tac (fun _ _ -> str "whole_start") (whole_start tac_end nb_args is_mes fonctional_ref
                                   input_type relation rec_arg_num ))) pstate
  in
  let pstate = start_proof Global.(env ()) ctx tclIDTAC tclIDTAC in
  try
    let sigma, new_goal_type = build_new_goal_type pstate in
    let sigma = Evd.from_ctx (Evd.evar_universe_context sigma) in
    open_new_goal pstate start_proof sigma
      using_lemmas tcc_lemma_ref
      (Some tcc_lemma_name)
      (new_goal_type)
  with EmptySubgoals ->
    (* a non recursive function declared with measure ! *)
    tcc_lemma_ref := Not_needed;
    defined pstate

let start_equation (f:GlobRef.t) (term_f:GlobRef.t)
  (cont_tactic:Id.t list -> tactic) g =
  let sigma = project g in
  let ids = pf_ids_of_hyps g in
  let terminate_constr = constr_of_global term_f in
  let terminate_constr = EConstr.of_constr terminate_constr in
  let nargs = nb_prod (project g) (EConstr.of_constr (type_of_const sigma terminate_constr)) in
  let x = n_x_id ids nargs in
  observe_tac (fun _ _ -> str "start_equation") (observe_tclTHENLIST (fun _ _ -> str "start_equation") [
    h_intros x;
    Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, evaluable_of_global_reference f)]);
    observe_tac (fun _ _ -> str "simplest_case")
      (Proofview.V82.of_tactic (simplest_case (mkApp (terminate_constr,
                             Array.of_list (List.map mkVar x)))));
    observe_tac (fun _ _ -> str "prove_eq") (cont_tactic x)]) g;;

let com_eqn sign uctx nb_arg eq_name functional_ref f_ref terminate_ref equation_lemma_type =
    let open CVars in
    let opacity =
      match terminate_ref with
| ConstRef c -> is_opaque_constant c
| _ -> anomaly ~label:"terminate_lemma" (Pp.str "not a constant.")
    in
    let evd = Evd.from_ctx uctx in
    let f_constr = constr_of_global f_ref in
    let equation_lemma_type = subst1 f_constr equation_lemma_type in
    let pstate = Lemmas.start_proof ~ontop:None eq_name (Global, false, Proof Lemma) ~sign evd
       (EConstr.of_constr equation_lemma_type) in
    let pstate = fst @@ by
       (Proofview.V82.tactic (start_equation f_ref terminate_ref
   (fun  x ->
      prove_eq (fun _ -> tclIDTAC)
        {nb_arg=nb_arg;
  f_terminate = EConstr.of_constr (constr_of_global terminate_ref);
         f_constr = EConstr.of_constr f_constr;
  concl_tac = tclIDTAC;
  func=functional_ref;
  info=(instantiate_lambda Evd.empty
          (EConstr.of_constr (def_of_const (constr_of_global functional_ref)))
          (EConstr.of_constr f_constr::List.map mkVar x)
  );
  is_main_branch = true;
  is_final = true;
  values_and_bounds = [];
  eqs = [];
  forbidden_ids = [];
  acc_inv = lazy (assert false);
  acc_id = Id.of_string "____";
  args_assoc = [];
  f_id = Id.of_string "______";
  rec_arg_id = Id.of_string "______";
  is_mes = false;
  ih = Id.of_string "______";
        }
   )
       )) pstate in
     (* (try Vernacentries.interp (Vernacexpr.VernacShow Vernacexpr.ShowProof) with _ -> ()); *)
(*      Vernacentries.interp (Vernacexpr.VernacShow Vernacexpr.ShowScript); *)
    let _ = Flags.silently (fun () -> Lemmas.save_proof_proved ?proof:None ~pstate ~opaque:opacity ~idopt:None) () in
    ()
(*      Pp.msgnl (fun _ _ -> str "eqn finished"); *)

let recursive_definition is_mes function_name rec_impls type_of_f r rec_arg_num eq
    generate_induction_principle using_lemmas : Proof_global.t option =
  let open Term in
  let open Constr in
  let open CVars in
  let env = Global.env() in
  let evd = Evd.from_env env in
  let evd, function_type = interp_type_evars ~program_mode:false env evd type_of_f in
  let function_r = Sorts.Relevant in (* TODO relevance *)
  let env = EConstr.push_named (Context.Named.Declaration.LocalAssum (make_annot function_name function_r,function_type)) env in
  (* Pp.msgnl (str "function type := " ++ Printer.pr_lconstr function_type);  *)
  let evd, ty = interp_type_evars ~program_mode:false env evd ~impls:rec_impls eq in
  let evd = Evd.minimize_universes evd in
  let equation_lemma_type = Reductionops.nf_betaiotazeta env evd (Evarutil.nf_evar evd ty) in
  let function_type = EConstr.to_constr ~abort_on_undefined_evars:false evd function_type in
  let equation_lemma_type = EConstr.Unsafe.to_constr equation_lemma_type in
(* Pp.msgnl (fun _ _ -> str "lemma type := " ++ Printer.pr_lconstr equation_lemma_type ++ fnl ()); *)
  let res_vars,eq' = decompose_prod equation_lemma_type in
  let env_eq' = Environ.push_rel_context (List.map (fun (x,y) -> LocalAssum (x,y)) res_vars) env in
  let eq' = Reductionops.nf_zeta env_eq' evd (EConstr.of_constr eq') in
  let eq' = EConstr.Unsafe.to_constr eq' in
  let res =
(*     Pp.msgnl (fun _ _ -> str "res_var :=" ++ Printer.pr_lconstr_env (push_rel_context (List.map (function (x,t) -> (x,None,t)) res_vars) env) eq'); *)
(*     Pp.msgnl (fun _ _ -> str "rec_arg_num := " ++ str (fun _ _ -> string_of_int rec_arg_num)); *)
(*     Pp.msgnl (fun _ _ -> str "eq' := " ++ str (fun _ _ -> string_of_int rec_arg_num)); *)
    match Constr.kind eq' with
      | App(e,[|_;_;eq_fix|]) ->
          mkLambda (make_annot (Name function_name) Sorts.Relevant,function_type,subst_var function_name (compose_lam res_vars  eq_fix))
      | _ -> failwith "Recursive Definition (res not eq)"
  in
  let pre_rec_args,function_type_before_rec_arg = decompose_prod_n (rec_arg_num - 1) function_type in
  let (_, rec_arg_type, _) = destProd function_type_before_rec_arg in
  let arg_types = List.rev_map snd (fst (decompose_prod_n (List.length res_vars) function_type)) in
  let equation_id = add_suffix function_name "_equation" in
  let functional_id =  add_suffix function_name "_F" in
  let term_id = add_suffix function_name "_terminate" in
  let functional_ref =
    let univs = Evd.univ_entry ~poly:false evd in
    declare_fun functional_id (IsDefinition Decl_kinds.Definition) ~univs res
  in
  (* Refresh the global universes, now including those of _F *)
  let evd = Evd.from_env (Global.env ()) in
  let env_with_pre_rec_args = push_rel_context(List.map (function (x,t) -> LocalAssum (x,t)) pre_rec_args) env in
  let relation, evuctx =
    interp_constr env_with_pre_rec_args evd r
  in
  let evd = Evd.from_ctx evuctx in
  let tcc_lemma_name = add_suffix function_name "_tcc" in
  let tcc_lemma_constr = ref Undefined in
  (* let _ = Pp.msgnl (fun _ _ -> str "relation := " ++ Printer.pr_lconstr_env env_with_pre_rec_args relation) in *)
  let hook uctx _ _ _ =
    let term_ref = Nametab.locate (qualid_of_ident term_id) in
    let f_ref = declare_f function_name (IsProof Lemma) arg_types term_ref in
    let _ = Extraction_plugin.Table.extraction_inline true [qualid_of_ident term_id] in
    (*     message "start second proof"; *)
    let stop =
      (* XXX: What is the correct way to get sign at hook time *)
      let sign = Environ.named_context_val Global.(env ()) in
      try com_eqn sign uctx (List.length res_vars) equation_id functional_ref f_ref term_ref (subst_var function_name equation_lemma_type);
   false
      with e when CErrors.noncritical e ->
begin
   if do_observe ()
   then Feedback.msg_debug (str "Cannot create equation Lemma " ++ CErrors.print e)
   else CErrors.user_err ~hdr:"Cannot create equation Lemma"
               (str "Cannot create equation lemma." ++ spc () ++
                str "This may be because the function is nested-recursive.")
   ;
   true
end
    in
    if not stop
    then
      let eq_ref = Nametab.locate (qualid_of_ident equation_id ) in
      let f_ref = destConst (constr_of_global f_ref)
      and functional_ref = destConst (constr_of_global functional_ref)
      and eq_ref = destConst (constr_of_global eq_ref) in
      generate_induction_principle f_ref tcc_lemma_constr
        functional_ref eq_ref rec_arg_num
        (EConstr.of_constr rec_arg_type)
        (nb_prod evd (EConstr.of_constr res)) relation;
      Flags.if_verbose
        msgnl (h 1 (Ppconstr.pr_id function_name ++
    spc () ++ str"is defined" )++ fnl () ++
      h 1 (Ppconstr.pr_id equation_id ++
      spc () ++ str"is defined" )
      )
  in
  (* XXX STATE Why do we need this... why is the toplevel protection not enough *)
  funind_purify (fun () ->
      let pstate = com_terminate
          tcc_lemma_name
          tcc_lemma_constr
          is_mes functional_ref
          (EConstr.of_constr rec_arg_type)
          relation rec_arg_num
          term_id
          using_lemmas
          (List.length res_vars)
          evd (Lemmas.mk_hook hook)
      in pstate) ()

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