(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(* $Id$ *)

open Pp
open Util
open Names
open Nameops
open Sign
open Term
open Termops
open Declarations
open Inductive
open Inductiveops
open Reductionops
open Environ
open Libnames
open Evd
open Pfedit
open Tacred
open Rawterm
open Tacmach
open Proof_trees
open Proof_type
open Logic
open Evar_refiner
open Clenv
open Clenvtac
open Refiner
open Tacticals
open Hipattern
open Coqlib
open Nametab
open Genarg
open Tacexpr
open Decl_kinds
open Evarutil
open Indrec

exception Bound

let rec nb_prod x =
  let rec count n c =
    match kind_of_term c with
        Prod(_,_,t) -> count (n+1) t
      | LetIn(_,a,_,t) -> count n (subst1 a t)
      | Cast(c,_,_) -> count n c
      | _ -> n
  in count 0 x

(*********************************************)
(*                 Tactics                   *)
(*********************************************)

(****************************************)
(* General functions                    *)
(****************************************)

(*
let get_pairs_from_bindings = 
  let pair_from_binding = function  
    | [(Bindings binds)] -> binds
    | _                  -> error "not a binding list!"
  in 
  List.map pair_from_binding
*)

let string_of_inductive c = 
  try match kind_of_term c with
  | Ind ind_sp -> 
      let (mib,mip) = Global.lookup_inductive ind_sp in 
      string_of_id mip.mind_typename
  | _ -> raise Bound
  with Bound -> error "Bound head variable"

let rec head_constr_bound t l =
  let t = strip_outer_cast(collapse_appl t) in
  match kind_of_term t with
    | Prod (_,_,c2)  -> head_constr_bound c2 l 
    | LetIn (_,_,_,c2) -> head_constr_bound c2 l 
    | App (f,args)  -> 
	head_constr_bound f (Array.fold_right (fun a l -> a::l) args l)
    | Const _        -> t::l
    | Ind _       -> t::l
    | Construct _ -> t::l
    | Var _          -> t::l
    | _                -> raise Bound

let head_constr c = 
  try head_constr_bound c [] with Bound -> error "Bound head variable"

(*
let bad_tactic_args s l =
  raise (RefinerError (BadTacticArgs (s,l)))
*)

(******************************************)
(*           Primitive tactics            *)
(******************************************)

let introduction    = Tacmach.introduction 
let intro_replacing = Tacmach.intro_replacing 
let internal_cut    = Tacmach.internal_cut
let internal_cut_rev = Tacmach.internal_cut_rev
let refine          = Tacmach.refine
let convert_concl   = Tacmach.convert_concl
let convert_hyp     = Tacmach.convert_hyp
let thin            = Tacmach.thin 
let thin_body       = Tacmach.thin_body

(* Moving hypotheses *)
let move_hyp        = Tacmach.move_hyp 

(* Renaming hypotheses *)
let rename_hyp      = Tacmach.rename_hyp

(* Refine as a fixpoint *)
let mutual_fix = Tacmach.mutual_fix

let fix ido n = match ido with
  | None -> mutual_fix (Pfedit.get_current_proof_name ()) n []
  | Some id -> mutual_fix id n []

(* Refine as a cofixpoint *)
let mutual_cofix = Tacmach.mutual_cofix

let cofix = function
  | None -> mutual_cofix (Pfedit.get_current_proof_name ()) []
  | Some id -> mutual_cofix id []

(**************************************************************)
(*          Reduction and conversion tactics                  *)
(**************************************************************)

type tactic_reduction = env -> evar_map -> constr -> constr

(* The following two tactics apply an arbitrary
   reduction function either to the conclusion or to a 
   certain hypothesis *)

let reduct_in_concl (redfun,sty) gl = 
  convert_concl_no_check (pf_reduce redfun gl (pf_concl gl)) sty gl

let reduct_in_hyp redfun (id,_,where) gl =
  let (_,c, ty) = pf_get_hyp gl id in
  let redfun' = (*under_casts*) (pf_reduce redfun gl) in
  match c with
  | None -> 
      if where = InHypValueOnly then
	errorlabstrm "" (pr_id id ++ str "has no value");
      convert_hyp_no_check (id,None,redfun' ty) gl
  | Some b ->
      let b' = if where <> InHypTypeOnly then redfun' b else b in
      let ty' =	if where <> InHypValueOnly then redfun' ty else ty in
      convert_hyp_no_check (id,Some b',ty') gl

let reduct_option redfun = function
  | Some id -> reduct_in_hyp (fst redfun) id 
  | None    -> reduct_in_concl redfun 

(* The following tactic determines whether the reduction
   function has to be applied to the conclusion or
   to the hypotheses. *) 

let redin_combinator redfun =
  onClauses (reduct_option redfun)

(* Now we introduce different instances of the previous tacticals *)
let change_and_check cv_pb t env sigma c =
  if is_fconv cv_pb env sigma t c then 
    t
  else 
    errorlabstrm "convert-check-hyp" (str "Not convertible")

(* Use cumulutavity only if changing the conclusion not a subterm *)
let change_on_subterm cv_pb t = function
  | None -> change_and_check cv_pb t
  | Some occl -> contextually false occl (change_and_check Reduction.CONV t) 

let change_in_concl occl t =
  reduct_in_concl ((change_on_subterm Reduction.CUMUL t occl),DEFAULTcast)

let change_in_hyp occl t   =
  reduct_in_hyp (change_on_subterm Reduction.CONV t occl)

let change_option occl t = function
    Some id -> change_in_hyp occl t id
  | None -> change_in_concl occl t

let change occl c cls =
  (match cls, occl with
      ({onhyps=(Some(_::_::_)|None)}|{onhyps=Some(_::_);onconcl=true}),
      Some _ ->
	error "No occurrences expected when changing several hypotheses"
    | _ -> ());
  onClauses (change_option occl c) cls

(* Pour usage interne (le niveau User est pris en compte par reduce) *)
let red_in_concl        = reduct_in_concl (red_product,DEFAULTcast)
let red_in_hyp          = reduct_in_hyp   red_product
let red_option          = reduct_option   (red_product,DEFAULTcast)
let hnf_in_concl        = reduct_in_concl (hnf_constr,DEFAULTcast)
let hnf_in_hyp          = reduct_in_hyp   hnf_constr
let hnf_option          = reduct_option   (hnf_constr,DEFAULTcast)
let simpl_in_concl      = reduct_in_concl (nf,DEFAULTcast)
let simpl_in_hyp        = reduct_in_hyp   nf
let simpl_option        = reduct_option   (nf,DEFAULTcast)
let normalise_in_concl  = reduct_in_concl (compute,DEFAULTcast)
let normalise_in_hyp    = reduct_in_hyp   compute
let normalise_option    = reduct_option   (compute,DEFAULTcast)
let normalise_vm_in_concl = reduct_in_concl (Redexpr.cbv_vm,VMcast)
let unfold_in_concl loccname = reduct_in_concl (unfoldn loccname,DEFAULTcast)
let unfold_in_hyp   loccname = reduct_in_hyp   (unfoldn loccname) 
let unfold_option   loccname = reduct_option (unfoldn loccname,DEFAULTcast) 
let pattern_option l = reduct_option (pattern_occs l,DEFAULTcast)

(* A function which reduces accordingly to a reduction expression,
   as the command Eval does. *)

let needs_check = function
  (* Expansion is not necessarily well-typed: e.g. expansion of t into x is
     not well-typed in [H:(P t); x:=t |- G] because x is defined after H *)
  | Fold _ -> true
  | _ -> false

let reduce redexp cl goal =
  (if needs_check redexp then with_check else (fun x -> x))
    (redin_combinator (Redexpr.reduction_of_red_expr redexp) cl)
    goal

(* Unfolding occurrences of a constant *)

let unfold_constr = function 
  | ConstRef sp -> unfold_in_concl [[],EvalConstRef sp]
  | VarRef id -> unfold_in_concl [[],EvalVarRef id]
  | _ -> errorlabstrm "unfold_constr" (str "Cannot unfold a non-constant.")

(*******************************************)
(*         Introduction tactics            *)
(*******************************************)

let fresh_id avoid id gl =
  next_global_ident_away true id (avoid@(pf_ids_of_hyps gl))

let id_of_name_with_default s = function
  | Anonymous -> id_of_string s
  | Name id   -> id

let default_id gl = function
  | (name,None,t) ->
      (match kind_of_term (pf_whd_betadeltaiota gl (pf_type_of gl t)) with
	| Sort (Prop _) -> (id_of_name_with_default "H" name)
	| Sort (Type _) -> (id_of_name_with_default "X" name)
	| _                   -> anomaly "Wrong sort")
  | (name,Some b,_) -> id_of_name_using_hdchar (pf_env gl) b name

(* Non primitive introduction tactics are treated by central_intro
   There is possibly renaming, with possibly names to avoid and 
   possibly a move to do after the introduction *)

type intro_name_flag =
  | IntroAvoid of identifier list
  | IntroBasedOn of identifier * identifier list
  | IntroMustBe of identifier

let find_name decl gl = function
  | IntroAvoid idl        -> 
      let id = fresh_id idl (default_id gl decl) gl in id
  | IntroBasedOn (id,idl) -> fresh_id idl id gl
  | IntroMustBe id        -> 
      let id' = fresh_id [] id gl in
      if id' <> id then error ((string_of_id id)^" is already used");
      id'

let build_intro_tac id = function
  | None      -> introduction id
  | Some dest -> tclTHEN (introduction id) (move_hyp true id dest)

let rec intro_gen name_flag move_flag force_flag gl =
  match kind_of_term (pf_concl gl) with
    | Prod (name,t,_) -> 
	build_intro_tac (find_name (name,None,t) gl name_flag) move_flag gl
    | LetIn (name,b,t,_) ->
	build_intro_tac (find_name (name,Some b,t) gl name_flag) move_flag gl
    | _ -> 
	if not force_flag then raise (RefinerError IntroNeedsProduct);
	try
	  tclTHEN
	    (reduce (Red true) onConcl)
	    (intro_gen name_flag move_flag force_flag) gl
	with Redelimination ->
	  errorlabstrm "Intro" (str "No product even after head-reduction")

let intro_mustbe_force id = intro_gen (IntroMustBe id) None true
let intro_using id = intro_gen (IntroBasedOn (id,[])) None false
let intro_force force_flag = intro_gen (IntroAvoid []) None force_flag
let intro = intro_force false
let introf = intro_force true

let introf_move_name destopt = intro_gen (IntroAvoid []) destopt true

(* For backwards compatibility *)
let central_intro = intro_gen

(**** Multiple introduction tactics ****)

let rec intros_using = function
    []      -> tclIDTAC
   | str::l  -> tclTHEN (intro_using str) (intros_using l)

let intros = tclREPEAT (intro_force false)

let intro_erasing id = tclTHEN (thin [id]) (introduction id)

let intros_replacing ids gls = 
  let rec introrec = function
    | [] -> tclIDTAC
    | id::tl ->
	(tclTHEN (tclORELSE (intro_replacing id)
		    (tclORELSE (intro_erasing id)   (* ?? *)
                       (intro_using id)))
           (introrec tl))
  in 
  introrec ids gls

(* User-level introduction tactics *)

let intro_move idopt idopt' = match idopt with
  | None -> intro_gen (IntroAvoid []) idopt' true
  | Some id -> intro_gen (IntroMustBe id) idopt' true

let pf_lookup_hypothesis_as_renamed env ccl = function
  | AnonHyp n -> pf_lookup_index_as_renamed env ccl n
  | NamedHyp id -> pf_lookup_name_as_renamed env ccl id

let pf_lookup_hypothesis_as_renamed_gen red h gl =
  let env = pf_env gl in
  let rec aux ccl =
    match pf_lookup_hypothesis_as_renamed env ccl h with
      | None when red ->
          aux 
	    ((fst (Redexpr.reduction_of_red_expr (Red true))) 
	       env (project gl) ccl)
      | x -> x
  in
  try aux (pf_concl gl)
  with Redelimination -> None

let is_quantified_hypothesis id g =
  match pf_lookup_hypothesis_as_renamed_gen true (NamedHyp id) g with
    | Some _ -> true
    | None -> false

let msg_quantified_hypothesis = function
  | NamedHyp id -> 
      str "hypothesis " ++ pr_id id
  | AnonHyp n ->
      int n ++ str (match n with 1 -> "st" | 2 -> "nd" | _ -> "th") ++
      str " non dependent hypothesis"

let depth_of_quantified_hypothesis red h gl =
  match pf_lookup_hypothesis_as_renamed_gen red h gl with
    | Some depth -> depth
    | None ->
        errorlabstrm "lookup_quantified_hypothesis" 
          (str "No " ++ msg_quantified_hypothesis h ++
	  str " in current goal" ++
	  if red then str " even after head-reduction" else mt ())

let intros_until_gen red h g =
  tclDO (depth_of_quantified_hypothesis red h g) intro g

let intros_until_id id = intros_until_gen true (NamedHyp id)
let intros_until_n_gen red n = intros_until_gen red (AnonHyp n)

let intros_until = intros_until_gen true
let intros_until_n = intros_until_n_gen true
let intros_until_n_wored = intros_until_n_gen false

let try_intros_until tac = function
  | NamedHyp id -> tclTHEN (tclTRY (intros_until_id id)) (tac id)
  | AnonHyp n -> tclTHEN (intros_until_n n) (onLastHyp tac)

let rec intros_move = function
  | [] -> tclIDTAC
  | (hyp,destopt) :: rest ->
      tclTHEN (intro_gen (IntroMustBe hyp) destopt false)
	(intros_move rest)

let dependent_in_decl a (_,c,t) =
  match c with
    | None -> dependent a t
    | Some body -> dependent a body || dependent a t

let move_to_rhyp rhyp gl =
  let rec get_lhyp lastfixed depdecls = function
    | [] ->
	(match rhyp with
	   | None -> lastfixed
      	   | Some h -> anomaly ("Hypothesis should occur: "^ (string_of_id h)))
    | (hyp,c,typ) as ht :: rest ->
	if Some hyp = rhyp then 
	  lastfixed
	else if List.exists (occur_var_in_decl (pf_env gl) hyp) depdecls then 
	  get_lhyp lastfixed (ht::depdecls) rest
        else
	  get_lhyp (Some hyp) depdecls rest
  in
  let sign = pf_hyps gl in
  let (hyp,c,typ as decl) = List.hd sign in
  match get_lhyp None [decl] (List.tl sign) with
    | None -> tclIDTAC gl
    | Some hypto -> move_hyp true hyp hypto gl

let rec intros_rmove = function
  | [] -> tclIDTAC
  | (hyp,destopt) :: rest ->
      tclTHENLIST [ introduction hyp;
 		    move_to_rhyp destopt;
		    intros_rmove rest ]

(****************************************************)
(*            Resolution tactics                    *)
(****************************************************)

(*  Refinement tactic: unification with the head of the head normal form
 *  of the type of a term. *)

let apply_type hdcty argl gl =
  refine (applist (mkCast (Evarutil.mk_new_meta(),DEFAULTcast, hdcty),argl)) gl
    
let apply_term hdc argl gl =
  refine (applist (hdc,argl)) gl

let bring_hyps hyps = 
  if hyps = [] then Refiner.tclIDTAC
  else
    (fun gl ->
      let newcl = List.fold_right mkNamedProd_or_LetIn hyps (pf_concl gl) in
      let f = mkCast (Evarutil.mk_new_meta(),DEFAULTcast, newcl) in
      refine_no_check (mkApp (f, instance_from_named_context hyps)) gl)

(* Resolution with missing arguments *)

let apply_with_bindings (c,lbind) gl = 
  (* The actual type of the theorem. It will be matched against the
  goal. If this fails, then the head constant will be unfolded step by
  step. *)
  let thm_ty0 = nf_betaiota (pf_type_of gl c) in
  let rec try_apply thm_ty =
    try
      let n = nb_prod thm_ty - nb_prod (pf_concl gl) in
      if n<0 then error "Apply: theorem has not enough premisses.";
      let clause = make_clenv_binding_apply gl n (c,thm_ty) lbind in
      Clenvtac.res_pf clause gl
    with (Pretype_errors.PretypeError _|RefinerError _|UserError _|Failure _) as exn ->
      let red_thm =
        try red_product (pf_env gl) (project gl) thm_ty
        with (Redelimination | UserError _) -> raise exn in
      try_apply red_thm in
  try try_apply thm_ty0
  with (Pretype_errors.PretypeError _|RefinerError _|UserError _|Failure _) ->
    (* Last chance: if the head is a variable, apply may try
       second order unification *)
    let clause = make_clenv_binding_apply gl (-1) (c,thm_ty0) lbind in 
    Clenvtac.res_pf clause gl

let apply c = apply_with_bindings (c,NoBindings)

let apply_list = function 
  | c::l -> apply_with_bindings (c,ImplicitBindings l)
  | _ -> assert false

(* Resolution with no reduction on the type *)

let apply_without_reduce c gl = 
  let clause = mk_clenv_type_of gl c in 
  res_pf clause gl

(* A useful resolution tactic which, if c:A->B, transforms |- C into
   |- B -> C and |- A

   -------------------
   Gamma |- c : A -> B      Gamma |- ?2 : A
   ----------------------------------------
           Gamma |- B                        Gamma |- ?1 : B -> C
           -----------------------------------------------------
                             Gamma |- ? : C

 Ltac lapply c :=
  let ty := check c in
  match eval hnf in ty with
    ?A -> ?B => cut B; [ idtac | apply c ]
  end.
*)

(**************************)
(*     Cut tactics        *)
(**************************)

let cut_and_apply c gl =
  let goal_constr = pf_concl gl in 
  match kind_of_term (pf_hnf_constr gl (pf_type_of gl c)) with
    | Prod (_,c1,c2) when not (dependent (mkRel 1) c2) ->
	tclTHENLAST
	  (apply_type (mkProd (Anonymous,c2,goal_constr)) [mkMeta(new_meta())])
	  (apply_term c [mkMeta (new_meta())]) gl
    | _ -> error "Imp_elim needs a non-dependent product"

let cut c gl =
  match kind_of_term (hnf_type_of gl c) with
    | Sort _ ->
        let id=next_name_away_with_default "H" Anonymous (pf_ids_of_hyps gl) in
        let t = mkProd (Anonymous, c, pf_concl gl) in
        tclTHENFIRST
          (internal_cut_rev id c)
          (tclTHEN (apply_type t [mkVar id]) (thin [id]))
          gl
    | _  -> error "Not a proposition or a type"

let cut_intro t = tclTHENFIRST (cut t) intro

let cut_replacing id t tac = 
  tclTHENS (cut t)
    [tclORELSE
      (intro_replacing id) 
      (tclORELSE (intro_erasing id) (intro_using id));
     tac (refine_no_check (mkVar id)) ]

let cut_in_parallel l = 
  let rec prec = function
    | [] -> tclIDTAC 
    | h::t -> tclTHENFIRST (cut h) (prec t)
  in 
  prec (List.rev l)

(********************************************************************)
(*               Exact tactics                                      *)
(********************************************************************)

let exact_check c gl =
  let concl = (pf_concl gl) in
  let ct = pf_type_of gl c in
  if pf_conv_x_leq gl ct concl then  
    refine_no_check c gl 
  else 
    error "Not an exact proof"

let exact_no_check = refine_no_check

let exact_proof c gl =
  (* on experimente la synthese d'ise dans exact *)
  let c = Constrintern.interp_casted_constr (project gl) (pf_env gl) c (pf_concl gl)
  in refine_no_check c gl 

let (assumption : tactic) = fun gl ->
  let concl =  pf_concl gl in 
  let hyps = pf_hyps gl in
  let rec arec only_eq = function
    | [] -> 
        if only_eq then arec false hyps else error "No such assumption"
    | (id,c,t)::rest -> 
	if (only_eq & eq_constr t concl) 
          or (not only_eq & pf_conv_x_leq gl t concl)
        then refine_no_check (mkVar id) gl
	else arec only_eq rest
  in
  arec true hyps

(*****************************************************************)
(*          Modification of a local context                      *)
(*****************************************************************)

(* This tactic enables the user to remove hypotheses from the signature.
 * Some care is taken to prevent him from removing variables that are 
 * subsequently used in other hypotheses or in the conclusion of the  
 * goal. *)                                                               

let clear ids gl = (* avant seul dyn_clear n'echouait pas en [] *)
  if ids=[] then tclIDTAC gl else with_check (thin ids) gl

let clear_body = thin_body

(*   Takes a list of booleans, and introduces all the variables 
 *  quantified in the goal which are associated with a value
 *  true  in the boolean list. *)

let rec intros_clearing = function
  | []          -> tclIDTAC
  | (false::tl) -> tclTHEN intro (intros_clearing tl)
  | (true::tl)  ->
      tclTHENLIST
        [ intro; onLastHyp (fun id -> clear [id]); intros_clearing tl]

(* Adding new hypotheses  *)

let new_hyp mopt (c,lbind) g =
  let clause  = make_clenv_binding g (c,pf_type_of g c) lbind in
  let (thd,tstack) = whd_stack (clenv_value clause) in
  let nargs = List.length tstack in
  let cut_pf = 
    applist(thd, 
            match mopt with
	      | Some m -> if m < nargs then list_firstn m tstack else tstack
	      | None   -> tstack)
  in 
  (tclTHENLAST (tclTHEN (tclEVARS (evars_of clause.env))
               (cut (pf_type_of g cut_pf)))
     ((tclORELSE (apply cut_pf) (exact_no_check cut_pf)))) g

(* Keeping only a few hypotheses *)

let keep hyps gl =
  let env = Global.env() in
  let ccl = pf_concl gl in
  let cl,_ =
    fold_named_context_reverse (fun (clear,keep) (hyp,_,_ as decl) ->
      if List.mem hyp hyps
	or List.exists (occur_var_in_decl env hyp) keep
	or occur_var env hyp ccl
      then (clear,decl::keep)
      else (hyp::clear,keep))
      ~init:([],[]) (pf_env gl)
  in thin cl gl

(************************)
(* Introduction tactics *)
(************************)

let constructor_tac boundopt i lbind gl =
  let cl = pf_concl gl in 
  let (mind,redcl) = pf_reduce_to_quantified_ind gl cl in 
  let nconstr =
    Array.length (snd (Global.lookup_inductive mind)).mind_consnames in
  if i=0 then error "The constructors are numbered starting from 1";
  if i > nconstr then error "Not enough constructors";
  begin match boundopt with 
    | Some expctdnum -> 
        if expctdnum <> nconstr then 
	  error "Not the expected number of constructors"
    | None -> ()
  end;
  let cons = mkConstruct (ith_constructor_of_inductive mind i) in
  let apply_tac = apply_with_bindings (cons,lbind) in
  (tclTHENLIST 
     [convert_concl_no_check redcl DEFAULTcast; intros; apply_tac]) gl

let one_constructor i = constructor_tac None i

(* Try to apply the constructor of the inductive definition followed by 
   a tactic t given as an argument.
   Should be generalize in Constructor (Fun c : I -> tactic)
 *)

let any_constructor tacopt gl =
  let t = match tacopt with None -> tclIDTAC | Some t -> t in
  let mind = fst (pf_reduce_to_quantified_ind gl (pf_concl gl)) in
  let nconstr =
    Array.length (snd (Global.lookup_inductive mind)).mind_consnames in
  if nconstr = 0 then error "The type has no constructors";
  tclFIRST (List.map (fun i -> tclTHEN (one_constructor i NoBindings) t) 
              (interval 1 nconstr)) gl

let left           = constructor_tac (Some 2) 1
let simplest_left  = left NoBindings

let right          = constructor_tac (Some 2) 2
let simplest_right = right NoBindings

let split          = constructor_tac (Some 1) 1
let simplest_split = split NoBindings

(********************************************)
(*       Elimination tactics                *)
(********************************************)

let last_arg c = match kind_of_term c with
  | App (f,cl) ->  
      array_last cl
  | _ -> anomaly "last_arg"
	
let elimination_clause_scheme allow_K elimclause indclause gl = 
  let indmv = 
    (match kind_of_term (last_arg elimclause.templval.rebus) with
       | Meta mv -> mv
       | _  -> errorlabstrm "elimination_clause"
             (str "The type of elimination clause is not well-formed")) 
  in
  let elimclause' = clenv_fchain indmv elimclause indclause in 
  res_pf elimclause' ~allow_K:allow_K gl

(* cast added otherwise tactics Case (n1,n2) generates (?f x y) and 
 * refine fails *)

let type_clenv_binding wc (c,t) lbind = 
  clenv_type (make_clenv_binding wc (c,t) lbind)

(* 
 * Elimination tactic with bindings and using an arbitrary 
 * elimination constant called elimc. This constant should end 
 * with a clause (x:I)(P .. ), where P is a bound variable.
 * The term c is of type t, which is a product ending with a type 
 * matching I, lbindc are the expected terms for c arguments 
 *)

let general_elim_clause elimtac (c,lbindc) (elimc,lbindelimc) gl =
  let ct = pf_type_of gl c in
  let t = try snd (pf_reduce_to_quantified_ind gl ct) with UserError _ -> ct in
  let indclause  = make_clenv_binding gl (c,t) lbindc  in
  let elimt      = pf_type_of gl elimc in
  let elimclause = make_clenv_binding gl (elimc,elimt) lbindelimc in 
  elimtac elimclause indclause gl

let general_elim c e ?(allow_K=true) =
  general_elim_clause (elimination_clause_scheme allow_K) c e

(* Elimination tactic with bindings but using the default elimination 
 * constant associated with the type. *)

let find_eliminator c gl =
  let (ind,t) = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in
  lookup_eliminator ind (elimination_sort_of_goal gl)

let default_elim (c,_ as cx) gl = 
  general_elim cx (find_eliminator c gl,NoBindings) gl

let elim_in_context c = function
  | Some elim -> general_elim c elim ~allow_K:true
  | None -> default_elim c

let elim (c,lbindc as cx) elim =
  match kind_of_term c with
    | Var id when lbindc = NoBindings ->
	tclTHEN (tclTRY (intros_until_id id)) (elim_in_context cx elim)
    | _ -> elim_in_context cx elim

(* The simplest elimination tactic, with no substitutions at all. *)

let simplest_elim c = default_elim (c,NoBindings)

(* Elimination in hypothesis *)

let elimination_in_clause_scheme id elimclause indclause gl =
  let (hypmv,indmv) = 
    match clenv_independent elimclause with
        [k1;k2] -> (k1,k2)
      | _  -> errorlabstrm "elimination_clause"
          (str "The type of elimination clause is not well-formed") in
  let elimclause'  = clenv_fchain indmv elimclause indclause in 
  let hyp = mkVar id in
  let hyp_typ = pf_type_of gl hyp in
  let hypclause =
    mk_clenv_from_n gl (Some 0) (hyp, hyp_typ) in
  let elimclause'' = clenv_fchain hypmv elimclause' hypclause in  
  let new_hyp_prf  = clenv_value elimclause'' in
  let new_hyp_typ  = clenv_type elimclause'' in
  if eq_constr hyp_typ new_hyp_typ then
    errorlabstrm "general_rewrite_in" 
      (str "Nothing to rewrite in " ++ pr_id id);
  tclTHEN
    (tclEVARS (evars_of elimclause''.env))
    (cut_replacing id new_hyp_typ
      (fun x gls -> refine_no_check new_hyp_prf gls)) gl

let general_elim_in id =
  general_elim_clause (elimination_in_clause_scheme id)

(* Case analysis tactics *)

let general_case_analysis_in_context (c,lbindc) gl =
  let (mind,_) = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in
  let sort     = elimination_sort_of_goal gl in
  let case = 
    if occur_term c (pf_concl gl) then make_case_dep else make_case_gen in
  let elim     = pf_apply case gl mind sort in 
  general_elim (c,lbindc) (elim,NoBindings) gl

let general_case_analysis (c,lbindc as cx) =
  match kind_of_term c with
    | Var id when lbindc = NoBindings ->
	tclTHEN (tclTRY (intros_until_id id))
	(general_case_analysis_in_context cx)
    | _ ->
	general_case_analysis_in_context cx

let simplest_case c = general_case_analysis (c,NoBindings)

(*****************************)
(* Decomposing introductions *)
(*****************************)

let clear_last = tclLAST_HYP (fun c -> (clear [destVar c]))
let case_last  = tclLAST_HYP simplest_case

let rec explicit_intro_names = function
| (IntroWildcard | IntroAnonymous) :: l -> explicit_intro_names l
| IntroIdentifier id :: l -> id :: explicit_intro_names l
| IntroOrAndPattern ll :: l' -> 
    List.flatten (List.map (fun l -> explicit_intro_names (l@l')) ll)
| [] -> []

  (* We delay thinning until the completion of the whole intros tactic
     to ensure that dependent hypotheses are cleared in the right
     dependency order (see bug #1000); we use fresh names, not used in
     the tactic, for the hyps to clear *)
let rec intros_patterns avoid thin destopt = function
  | IntroWildcard :: l ->
      tclTHEN 
        (intro_gen (IntroAvoid (avoid@explicit_intro_names l)) None true)
        (onLastHyp (fun id ->
	  tclORELSE
	    (tclTHEN (clear [id]) (intros_patterns avoid thin destopt l))
	    (intros_patterns avoid (id::thin) destopt l)))
  | IntroIdentifier id :: l ->
      tclTHEN
        (intro_gen (IntroMustBe id) destopt true)
        (intros_patterns avoid thin destopt l)
  | IntroAnonymous :: l ->
      tclTHEN
        (intro_gen (IntroAvoid (avoid@explicit_intro_names l)) destopt true)
        (intros_patterns avoid thin destopt l)
  | IntroOrAndPattern ll :: l' ->
      tclTHEN
        introf
        (tclTHENS
	  (tclTHEN case_last clear_last)
	  (List.map (fun l -> intros_patterns avoid thin destopt (l@l')) ll))
  | [] -> clear thin

let intros_pattern = intros_patterns [] []

let intro_pattern destopt pat = intros_patterns [] [] destopt [pat]

let intro_patterns = function 
  | [] -> tclREPEAT intro
  | l  -> intros_pattern None l

(**************************)
(*   Other cut tactics    *)
(**************************)

let hid = id_of_string "H"
let xid = id_of_string "X"

let make_id s = fresh_id [] (match s with Prop _ -> hid | Type _ -> xid)

let prepare_intros s ipat gl = match ipat with
  | IntroAnonymous -> make_id s gl, tclIDTAC
  | IntroWildcard -> let id = make_id s gl in id, thin [id]
  | IntroIdentifier id -> id, tclIDTAC
  | IntroOrAndPattern ll -> make_id s gl, 
    (tclTHENS 
      (tclTHEN case_last clear_last)
      (List.map (intros_pattern None) ll))

let ipat_of_name = function
  | Anonymous -> IntroAnonymous
  | Name id -> IntroIdentifier id

let assert_as first ipat c gl =
  match kind_of_term (hnf_type_of gl c) with
  | Sort s ->
      let id,tac = prepare_intros s ipat gl in
      tclTHENS ((if first then internal_cut else internal_cut_rev) id c)
	(if first then [tclIDTAC; tac] else [tac; tclIDTAC]) gl
  | _  -> error "Not a proposition or a type"

let assert_tac first na = assert_as first (ipat_of_name na)
let true_cut = assert_tac true 

(**************************)
(*   Generalize tactics   *)
(**************************)

let generalize_goal gl c cl =
  let t = pf_type_of gl c in
  match kind_of_term c with
    | Var id ->
	(* The choice of remembering or not a non dependent name has an impact
	   on the future Intro naming strategy! *)
	(* if dependent c cl then mkNamedProd id t cl
	   else mkProd (Anonymous,t,cl) *)
	mkNamedProd id t cl
    | _        -> 
        let cl' = subst_term c cl in
        if noccurn 1 cl' then 
	  mkProd (Anonymous,t,cl)
          (* On ne se casse pas la tete : on prend pour nom de variable
             la premiere lettre du type, meme si "ci" est une
             constante et qu'on pourrait prendre directement son nom *)
        else 
	  prod_name (Global.env()) (Anonymous, t, cl')

let generalize_dep c gl =
  let env = pf_env gl in
  let sign = pf_hyps gl in
  let init_ids = ids_of_named_context (Global.named_context()) in
  let rec seek d toquant =
    if List.exists (fun (id,_,_) -> occur_var_in_decl env id d) toquant
      or dependent_in_decl c d then 
      d::toquant
    else 
      toquant in
  let to_quantify = Sign.fold_named_context seek sign ~init:[] in
  let to_quantify_rev = List.rev to_quantify in
  let qhyps = List.map (fun (id,_,_) -> id) to_quantify_rev in
  let tothin = List.filter (fun id -> not (List.mem id init_ids)) qhyps in
  let tothin' =
    match kind_of_term c with
      | Var id when mem_named_context id sign & not (List.mem id init_ids)
	  -> id::tothin
      | _ -> tothin
  in
  let cl' = it_mkNamedProd_or_LetIn (pf_concl gl) to_quantify in
  let cl'' = generalize_goal gl c cl' in
  let args = Array.to_list (instance_from_named_context to_quantify_rev) in
  tclTHEN
    (apply_type cl'' (c::args))
    (thin (List.rev tothin'))
    gl
    
let generalize lconstr gl = 
  let newcl = List.fold_right (generalize_goal gl) lconstr (pf_concl gl) in
  apply_type newcl lconstr gl

(* Faudra-t-il une version avec plusieurs args de generalize_dep ?
Cela peut-être troublant de faire "Generalize Dependent H n" dans
"n:nat; H:n=n |- P(n)" et d'échouer parce que H a disparu après la
généralisation dépendante par n.

let quantify lconstr =
 List.fold_right 
   (fun com tac -> tclTHEN tac (tactic_com generalize_dep c))
   lconstr
   tclIDTAC
*)

(* A dependent cut rule à la sequent calculus
   ------------------------------------------
   Sera simplifiable le jour où il y aura un let in primitif dans constr

   [letin_tac b na c (occ_hyp,occ_ccl) gl] transforms
   [...x1:T1(c),...,x2:T2(c),... |- G(c)] into
   [...x:T;x1:T1(x),...,x2:T2(x),... |- G(x)] if [b] is false or
   [...x:=c:T;x1:T1(x),...,x2:T2(x),... |- G(x)] if [b] is true

   [occ_hyp,occ_ccl] tells which occurrences of [c] have to be substituted;
   if [occ_hyp = []] and [occ_ccl = None] then [c] is substituted
   wherever it occurs, otherwise [c] is substituted only in hyps
   present in [occ_hyps] at the specified occurrences (everywhere if
   the list of occurrences is empty), and in the goal at the specified
   occurrences if [occ_goal] is not [None];

   if name = Anonymous, the name is build from the first letter of the type;

   The tactic first quantify the goal over x1, x2,... then substitute then
   re-intro x1, x2,... at their initial place ([marks] is internally
   used to remember the place of x1, x2, ...: it is the list of hypotheses on
   the left of each x1, ...).
*)



let occurrences_of_hyp id cls =
  let rec hyp_occ = function
      [] -> None
    | (id',occs,hl)::_ when id=id' -> Some occs
    | _::l -> hyp_occ l in
  match cls.onhyps with
      None -> Some []
    | Some l -> hyp_occ l

let occurrences_of_goal cls =
  if cls.onconcl then Some cls.concl_occs else None

let in_every_hyp cls = (cls.onhyps=None)

(*
(* Implementation with generalisation then re-intro: introduces noise *)
(* in proofs *)

let letin_abstract id c occs gl =
  let env = pf_env gl in
  let compute_dependency _ (hyp,_,_ as d) ctxt =
    let d' =
      try
	match occurrences_of_hyp hyp occs with
	  | None -> raise Not_found
	  | Some occ ->
              let newdecl = subst_term_occ_decl occ c d in
              if occ = [] & d = newdecl then
		if not (in_every_hyp occs)
		then raise (RefinerError (DoesNotOccurIn (c,hyp)))
		else raise Not_found
              else 
		(subst1_decl (mkVar id) newdecl, true)
	with Not_found -> 
	  (d,List.exists
	      (fun ((id,_,_),dep) -> dep && occur_var_in_decl env id d) ctxt)
    in d'::ctxt
  in 
  let ctxt' = fold_named_context compute_dependency env ~init:[] in
  let compute_marks ((depdecls,marks as accu),lhyp) ((hyp,_,_) as d,b) =
    if b then ((d::depdecls,(hyp,lhyp)::marks), lhyp)
    else (accu, Some hyp) in
  let (depdecls,marks),_ = List.fold_left compute_marks (([],[]),None) ctxt' in
  let ccl = match occurrences_of_goal occs with
    | None -> pf_concl gl
    | Some occ -> subst1 (mkVar id) (subst_term_occ occ c (pf_concl gl))
  in
  (depdecls,marks,ccl)

let letin_tac with_eq name c occs gl =
  let x = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c) name in
  let id =
    if name = Anonymous then fresh_id [] x gl else
      if not (mem_named_context x (pf_hyps gl)) then x else
	error ("The variable "^(string_of_id x)^" is already declared") in
  let (depdecls,marks,ccl)= letin_abstract id c occs gl in 
  let t = pf_type_of gl c in
  let tmpcl = List.fold_right mkNamedProd_or_LetIn depdecls ccl in
  let args = Array.to_list (instance_from_named_context depdecls) in
  let newcl = mkNamedLetIn id c t tmpcl in
  let lastlhyp = if marks=[] then None else snd (List.hd marks) in
  tclTHENLIST
    [ apply_type newcl args;
      thin (List.map (fun (id,_,_) -> id) depdecls);
      intro_gen (IntroMustBe id) lastlhyp false;
      if with_eq then tclIDTAC else thin_body [id];
      intros_move marks ] gl
*)

(* Implementation without generalisation: abbrev will be lost in hyps in *)
(* in the extracted proof *)

let letin_abstract id c occs gl =
  let env = pf_env gl in
  let compute_dependency _ (hyp,_,_ as d) depdecls =
    match occurrences_of_hyp hyp occs with
      | None -> depdecls
      | Some occ ->
          let newdecl = subst_term_occ_decl occ c d in
          if occ = [] & d = newdecl then
	    if not (in_every_hyp occs)
	    then raise (RefinerError (DoesNotOccurIn (c,hyp)))
	    else depdecls
          else 
	    (subst1_decl (mkVar id) newdecl)::depdecls in 
  let depdecls = fold_named_context compute_dependency env ~init:[] in
  let ccl = match occurrences_of_goal occs with
    | None -> pf_concl gl
    | Some occ -> subst1 (mkVar id) (subst_term_occ occ c (pf_concl gl)) in
  let lastlhyp = if depdecls = [] then None else Some(pi1(list_last depdecls)) in
  (depdecls,lastlhyp,ccl)

let letin_tac with_eq name c occs gl =
  let id =
    let x = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c) name in
    if name = Anonymous then fresh_id [] x gl else
      if not (mem_named_context x (pf_hyps gl)) then x else
	error ("The variable "^(string_of_id x)^" is already declared") in
  let (depdecls,lastlhyp,ccl)= letin_abstract id c occs gl in 
  let t = refresh_universes (pf_type_of gl c) in
  let newcl = mkNamedLetIn id c t ccl in
  tclTHENLIST
    [ convert_concl_no_check newcl DEFAULTcast;
      intro_gen (IntroMustBe id) lastlhyp true;
      if with_eq then tclIDTAC else thin_body [id];
      tclMAP convert_hyp_no_check depdecls ] gl
  
(* Tactics "pose proof" (usetac=None) and "assert" (otherwise) *)
let forward usetac ipat c gl =
  match usetac with
  | None -> 
      let t = refresh_universes (pf_type_of gl c) in
      tclTHENS (assert_as true ipat t) [exact_no_check c; tclIDTAC] gl
  | Some tac -> 
      tclTHENS (assert_as true ipat c) [tac; tclIDTAC] gl

(*****************************)
(* High-level induction      *)
(*****************************)

(*
 * A "natural" induction tactic
 * 
  - [H0:T0, ..., Hi:Ti, hyp0:P->I(args), Hi+1:Ti+1, ..., Hn:Tn |-G] is the goal
  - [hyp0] is the induction hypothesis
  - we extract from [args] the variables which are not rigid parameters
    of the inductive type, this is [indvars] (other terms are forgotten);
    [indhyps] are the ones which actually are declared in context
    (done in [find_atomic_param_of_ind])
  - we look for all hyps depending of [hyp0] or one of [indvars]:
    this is [dephyps] of types [deptyps] respectively
  - [statuslist] tells for each hyps in [dephyps] after which other hyp
    fixed in the context they must be moved (when induction is done)
  - [hyp0succ] is the name of the hyp fixed in the context after which to
    move the subterms of [hyp0succ] in the i-th branch where it is supposed
    to be the i-th constructor of the inductive type.

  Strategy: (cf in [induction_from_context])
  - requantify and clear all [dephyps]
  - apply induction on [hyp0]
  - clear [indhyps] and [hyp0]
  - in the i-th subgoal, intro the arguments of the i-th constructor
    of the inductive type after [hyp0succ] (done in
    [induct_discharge]) let the induction hypotheses on top of the
    hyps because they may depend on variables between [hyp0] and the
    top. A counterpart is that the dep hyps programmed to be intro-ed
    on top must now be intro-ed after the induction hypotheses
  - move each of [dephyps] at the right place following the
    [statuslist]

 *)

let check_unused_names names =
  if names <> [] & Options.is_verbose () then
    let s = if List.tl names = [] then " " else "s " in
    msg_warning 
      (str"Unused introduction pattern" ++ str s ++ 
       str": " ++ prlist_with_sep spc pr_intro_pattern names)

let rec first_name_buggy = function
  | IntroOrAndPattern [] -> None
  | IntroOrAndPattern ([]::l) -> first_name_buggy (IntroOrAndPattern l)
  | IntroOrAndPattern ((p::_)::_) -> first_name_buggy p
  | IntroWildcard -> None
  | IntroIdentifier id -> Some id
  | IntroAnonymous -> assert false

let consume_pattern avoid id gl = function
  | [] -> (IntroIdentifier (fresh_id avoid id gl), [])
  | IntroAnonymous::names ->
      let avoid = avoid@explicit_intro_names names in
      (IntroIdentifier (fresh_id avoid id gl), names)
  | pat::names -> (pat,names)

type elim_arg_kind = RecArg | IndArg | OtherArg

let induct_discharge statuslists destopt avoid' (avoid,ra) names gl =
  let avoid = avoid @ avoid' in
  let (lstatus,rstatus) = statuslists in
  let tophyp = ref None in
  let rec peel_tac ra names gl = match ra with
    | (RecArg,recvarname) ::
        (IndArg,hyprecname) :: ra' ->
        let recpat,names = match names with
          | [IntroIdentifier id as pat] ->
              let id = next_ident_away (add_prefix "IH" id) avoid in
	      (pat, [IntroIdentifier id])
          | _ -> consume_pattern avoid recvarname gl names in
        let hyprec,names = consume_pattern avoid hyprecname gl names in
	(* This is buggy for intro-or-patterns with different first hypnames *)
	if !tophyp=None then tophyp := first_name_buggy hyprec;
        tclTHENLIST
	  [ intros_patterns avoid [] destopt [recpat];
	    intros_patterns avoid [] None [hyprec];
	    peel_tac ra' names ] gl
    | (IndArg,hyprecname) :: ra' ->
	(* Rem: does not happen in Coq schemes, only in user-defined schemes *)
        let pat,names = consume_pattern avoid hyprecname gl names in
	tclTHEN (intros_patterns avoid [] destopt [pat]) (peel_tac ra' names) gl
    | (RecArg,recvarname) :: ra' ->
        let pat,names = consume_pattern avoid recvarname gl names in
	tclTHEN (intros_patterns avoid [] destopt [pat]) (peel_tac ra' names) gl
    | (OtherArg,_) :: ra' ->
        let pat,names = match names with
          | [] -> IntroAnonymous, []
          | pat::names -> pat,names in
	tclTHEN (intros_patterns avoid [] destopt [pat]) (peel_tac ra' names) gl
    | [] ->
        check_unused_names names;
        tclIDTAC gl
  in
  let intros_move lstatus =
    let newlstatus = (* if some IH has taken place at the top of hyps *)
      List.map (function (hyp,None) -> (hyp,!tophyp) | x -> x) lstatus in
    intros_move newlstatus
  in 
  tclTHENLIST [ peel_tac ra names;
		intros_rmove rstatus;
		intros_move lstatus ] gl

(* - le recalcul de indtyp à chaque itération de atomize_one est pour ne pas
     s'embêter à regarder si un letin_tac ne fait pas des
     substitutions aussi sur l'argument voisin *)

(* Marche pas... faut prendre en compte l'occurrence précise... *)

let atomize_param_of_ind (indref,nparams) hyp0 gl =
  let tmptyp0 = pf_get_hyp_typ gl hyp0 in
  let typ0 = pf_apply reduce_to_quantified_ref gl indref tmptyp0 in
  let prods, indtyp = decompose_prod typ0 in
  let argl = snd (decompose_app indtyp) in
  let params = list_firstn nparams argl in
  (* le gl est important pour ne pas préévaluer *)
  let rec atomize_one i avoid gl =
    if i<>nparams then
      let tmptyp0 = pf_get_hyp_typ gl hyp0 in
      (* If argl <> [], we expect typ0 not to be quantified, in order to
         avoid bound parameters... then we call pf_reduce_to_atomic_ind *)
      let indtyp = pf_apply reduce_to_atomic_ref gl indref tmptyp0 in
      let argl = snd (decompose_app indtyp) in
      let c = List.nth argl (i-1) in
      match kind_of_term c with
	| Var id when not (List.exists (occur_var (pf_env gl) id) avoid) ->
	    atomize_one (i-1) ((mkVar id)::avoid) gl
	| Var id ->
	    let x = fresh_id [] id gl in
	    tclTHEN
	      (letin_tac true (Name x) (mkVar id) allClauses)
	      (atomize_one (i-1) ((mkVar x)::avoid)) gl
	| _ ->
	    let id = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c)
		       Anonymous in
	    let x = fresh_id [] id gl in
	    tclTHEN
	      (letin_tac true (Name x) c allClauses)
	      (atomize_one (i-1) ((mkVar x)::avoid)) gl
    else 
      tclIDTAC gl
  in
  atomize_one (List.length argl) params gl

let find_atomic_param_of_ind nparams indtyp =
  let argl = snd (decompose_app indtyp) in
  let argv = Array.of_list argl in
  let params = list_firstn nparams argl in
  let indvars = ref Idset.empty in
  for i = nparams to (Array.length argv)-1 do
    match kind_of_term argv.(i) with
      | Var id
          when not (List.exists (occur_var (Global.env()) id) params) ->
	  indvars := Idset.add id !indvars
      | _ -> ()
  done;
  Idset.elements !indvars;
  

   (* [cook_sign] builds the lists [indhyps] of hyps that must be
   erased, the lists of hyps to be generalize [(hdeps,tdeps)] on the
   goal together with the places [(lstatus,rstatus)] where to re-intro
   them after induction. To know where to re-intro the dep hyp, we
   remember the name of the hypothesis [lhyp] after which (if the dep
   hyp is more recent than [hyp0]) or [rhyp] before which (if older
   than [hyp0]) its equivalent must be moved when the induction has
   been applied. Since computation of dependencies and [rhyp] is from
   more ancient (on the right) to more recent hyp (on the left) but
   the computation of [lhyp] progresses from the other way, [cook_hyp]
   is in two passes (an alternative would have been to write an
   higher-order algorithm). We strongly use references to reduce
   the accumulation of arguments.

   To summarize, the situation looks like this

   Goal(n,x) -| H6:(Q n); x:A; H5:True; H4:(le O n); H3:(P n); H2:True; n:nat
                Left                                                    Right 

   Induction hypothesis is H4 ([hyp0])
   Variable parameters of (le O n) is the singleton list with "n" ([indvars])
   Part of [indvars] really in context is the same ([indhyps])
   The dependent hyps are H3 and H6 ([dephyps])
   For H3 the memorized places are H5 ([lhyp]) and H2 ([rhyp])
    because these names are among the hyp which are fixed through the induction
   For H6 the neighbours are None ([lhyp]) and H5 ([rhyp])
   For H3, because on the right of H4, we remember rhyp (here H2)
   For H6, because on the left of H4, we remember lhyp (here None)
   For H4, we remember lhyp (here H5)

   The right neighbour is then translated into the left neighbour
   because move_hyp tactic needs the name of the hyp _after_ which we
   move the hyp to move.

   But, say in the 2nd subgoal of the hypotheses, the goal will be

   (m:nat)((P m)->(Q m)->(Goal m)) -> (P Sm)->   (Q Sm)->   (Goal Sm)
     ^^^^^^^^^^^^^^^^^^^^^^^^^^^       ^^^^
         both go where H4 was       goes where  goes where
                                      H3 was      H6 was

   We have to intro and move m and the recursive hyp first, but then
   where to move H3 ??? Only the hyp on its right is relevant, but we
   have to translate it into the name of the hyp on the left

   Note: this case where some hyp(s) in [dephyps] has(have) the same
   left neighbour as [hyp0] is the only problematic case with right
   neighbours. For the other cases (e.g. an hyp H1:(R n) between n and H2
   would have posed no problem. But for uniformity, we decided to use
   the right hyp for all hyps on the right of H4.

   Others solutions are welcome 

   PC 9 fev 06: Adapted to accept multi argument principle with no
   main arg hyp. hyp0 is now optional, meaning that it is possible
   that there is no main induction hypotheses. In this case, we
   consider the last "parameter" (in [indvars]) as the limit between
   "left" and "right", BUT it must be included in indhyps.

   Other solutions are still welcome

*)

exception Shunt of identifier option

let cook_sign hyp0_opt indvars_init env =
  let hyp0,indvars = 
    match hyp0_opt with
      | None -> List.hd (List.rev indvars_init) , indvars_init
      | Some h -> h,indvars_init in
  (* First phase from L to R: get [indhyps], [decldep] and [statuslist]
     for the hypotheses before (= more ancient than) hyp0 (see above) *)
  let allindhyps = hyp0::indvars in
  let indhyps = ref [] in
  let decldeps = ref [] in
  let ldeps = ref [] in
  let rstatus = ref [] in
  let lstatus = ref [] in
  let before = ref true in
  let seek_deps env (hyp,_,_ as decl) rhyp =
    if hyp = hyp0 then begin
      before:=false; 
      (* If there was no main induction hypotheses, then hyp is one of
         indvars too, so add it to indhyps. *)
      (if hyp0_opt=None then indhyps := hyp::!indhyps); 
      None (* fake value *)
    end else if List.mem hyp indvars then begin
      (* warning: hyp can still occur after induction *)
      (* e.g. if the goal (t hyp hyp0) with other occs of hyp in t *)
      indhyps := hyp::!indhyps; 
      rhyp
    end else
      if (List.exists (fun id -> occur_var_in_decl env id decl) allindhyps
	  or List.exists (fun (id,_,_) -> occur_var_in_decl env id decl)
        !decldeps)
      then begin
	decldeps := decl::!decldeps;
	if !before then 
	  rstatus := (hyp,rhyp)::!rstatus
	else 
	  ldeps := hyp::!ldeps; (* status computed in 2nd phase *)
	Some hyp end
      else
	Some hyp
  in
  let _ = fold_named_context seek_deps env ~init:None in
  (* 2nd phase from R to L: get left hyp of [hyp0] and [lhyps] *)
  let compute_lstatus lhyp (hyp,_,_) =
    if hyp = hyp0 then raise (Shunt lhyp);
    if List.mem hyp !ldeps then begin
      lstatus := (hyp,lhyp)::!lstatus;
      lhyp
    end else
      if List.mem hyp !indhyps then lhyp else (Some hyp) 
  in
  try 
    let _ = fold_named_context_reverse compute_lstatus ~init:None env in
(*     anomaly "hyp0 not found" *)
    raise (Shunt (None)) (* ?? FIXME *)
  with Shunt lhyp0 ->
    let statuslists = (!lstatus,List.rev !rstatus) in
    (statuslists, (if hyp0_opt=None then None else lhyp0) , !indhyps, !decldeps)


(* Unification between ((elimc:elimt) ?i ?j ?k ?l ... ?m) and the
   hypothesis on which the induction is made *)
let induction_tac varname typ ((elimc,lbindelimc),elimt) gl =
  let c = mkVar varname in
  let indclause  = make_clenv_binding gl (c,typ) NoBindings  in
  let elimclause =
    make_clenv_binding gl 
      (mkCast (elimc,DEFAULTcast, elimt),elimt) lbindelimc in
  elimination_clause_scheme true elimclause indclause gl

let make_base n id =
  if n=0 or n=1 then id
  else
    (* This extends the name to accept new digits if it already ends with *)
    (* digits *)
    id_of_string (atompart_of_id (make_ident (string_of_id id) (Some 0)))

(* Builds tw different names from an optional inductive type and a
   number, also deals with a list of names to avoid. If the inductive
   type is None, then hyprecname is HIi where i is a number. *)
let make_up_names n ind_opt cname = 
  let is_hyp = atompart_of_id cname = "H" in
  let base = string_of_id (make_base n cname) in
  let base_ind = 
    if is_hyp then 
      match ind_opt with
	| None -> id_of_string ""
	| Some ind_id -> Nametab.id_of_global ind_id 
    else cname in
  let hyprecname = add_prefix "IH" (make_base n base_ind) in
  let avoid =
    if n=1 (* Only one recursive argument *) or n=0 then []
    else
      (* Forbid to use cname, cname0, hyprecname and hyprecname0 *)
      (* in order to get names such as f1, f2, ... *)
      let avoid =
        (make_ident (string_of_id hyprecname) None) ::
        (make_ident (string_of_id hyprecname) (Some 0)) :: [] in
      if atompart_of_id cname <> "H" then
        (make_ident base (Some 0)) :: (make_ident base None) :: avoid
      else avoid in
  id_of_string base, hyprecname, avoid

let is_indhyp p n t =
  let l, c = decompose_prod t in
  let c,_ = decompose_app c in 
  let p = p + List.length l in
  match kind_of_term c with
    | Rel k when p < k & k <= p + n -> true
    | _ -> false

let chop_context n l = 
  let rec chop_aux acc = function
    | n, (_,Some _,_ as h :: t) -> chop_aux (h::acc) (n, t)
    | 0, l2 -> (List.rev acc, l2)
    | n, (h::t) -> chop_aux (h::acc) (n-1, t)
    | _, [] -> anomaly "chop_context"
  in 
  chop_aux [] (n,l)

let error_ind_scheme s =
  let s = if s <> "" then s^" " else s in
  error ("Cannot recognise "^s^"an induction schema")




let occur_rel n c = 
  let res = not (noccurn n c) in
  res

let list_filter_firsts f l =
  let rec list_filter_firsts_aux f acc l =
    match l with
      | e::l' when f e -> list_filter_firsts_aux f (acc@[e]) l'
      | _ -> acc,l
  in
  list_filter_firsts_aux f [] l

let count_rels_from n c =
  let rels = free_rels c in
  let cpt,rg = ref 0, ref n in
  while Intset.mem !rg rels do
    cpt:= !cpt+1; rg:= !rg+1;
  done;
  !cpt

let count_nonfree_rels_from n c =
  let rels = free_rels c in
  if Intset.exists (fun x -> x >= n) rels then
    let cpt,rg = ref 0, ref n in
    while not (Intset.mem !rg rels) do
      cpt:= !cpt+1; rg:= !rg+1;
    done;
    !cpt
  else raise Not_found


(* cuts a list in two parts, first of size n. Size must be greater than n *)
let cut_list n l =
  let rec cut_list_aux acc n l =
    if n<=0 then acc,l
    else match l with
      | [] -> assert false
      | e::l' -> cut_list_aux (acc@[e]) (n-1) l' in
  let res = cut_list_aux [] n l in
  res


(* This functions splits the products of the induction scheme [elimt] in three
   parts: 
    - branches, easily detectable (they are not referred by rels in the subterm)
    - what was found before branches (acc1) that is: parameters and predicates
    - what was found after branches (acc3) that is: args and indarg if any
   if there is no branch, we try to fill in acc3 with args/indargs.
   We also return the conclusion.
*)
let decompose_paramspred_branch_args elimt = 
  let rec cut_noccur elimt acc2 : rel_context * rel_context * types =
    match kind_of_term elimt with
      | Prod(nme,tpe,elimt')  -> 
	  let hd_tpe,_ = decompose_app 
	    (if isProd tpe then snd (decompose_prod tpe) else tpe) in
	  if not (occur_rel 1 elimt') && isRel hd_tpe	    
	  then cut_noccur elimt' ((nme,None,tpe)::acc2)
	  else 
	    let acc3,ccl = decompose_prod_assum elimt in
	    acc2 , acc3 , ccl
      | App(qi, args) -> acc2 , [] , elimt
      | Rel _ -> acc2 , [] , elimt
	  (* | LetIn ??*)
      | _ -> error "cannot recognise an induction schema" in

  let rec cut_occur elimt acc1 : rel_context * rel_context * rel_context * types =
    match kind_of_term elimt with
      | Prod(nme,tpe,elimt')  -> 
	  if occur_rel 1 elimt' then cut_occur elimt' ((nme,None,tpe)::acc1)
	  else (* cut_noccur will decide if this and the following are branches *)
	    let acc2,acc3,ccl = cut_noccur elimt [] in
	    acc1 , acc2 , acc3 , ccl
      | App(qi, args) -> acc1 , [] , [] , elimt
      | Rel _ -> acc1 , [] , [] , elimt
	  (* | LetIn ??*)
      | _ -> error "cannot recognise an induction schema" in

  let acc1, acc2 , acc3, ccl = cut_occur elimt [] in
  (* Particular treatment when dealing with a dependent empty type elim scheme:
     if there is no branch, then acc1 contains all hyps which is wrong (acc1
     should contain parameters and predicate only). This happens for an empty
     type (See for example Empty_set_ind, as False would actually be ok). Then
     we must find the predicate of the conclusion to separate params_pred from
     args. We suppose there is only one predicate here. *)
  if List.length acc2 <> 0 
  then acc1, acc2 , acc3, ccl
  else 
    let hyps,ccl = decompose_prod_assum elimt in
    let hd_ccl_pred,_ = decompose_app ccl in
    match kind_of_term hd_ccl_pred with
      | Rel i  -> 
	  let acc3,acc1 = cut_list (i-1) hyps in
	  acc1 , [] , acc3 , ccl
      | _ -> error "cannot recognize an induction schema"



let exchange_hd_app subst_hd t =
  let hd,args= decompose_app t in mkApp (subst_hd,Array.of_list args)

(*
   The general form of an induction principle is the following:
   
   forall prm1 prm2 ... prmp,                          (induction parameters)
   forall Q1...,(Qi:Ti_1 -> Ti_2 ->...-> Ti_ni),...Qq, (predicates)
   branch1, branch2, ... , branchr,                    (branches of the principle)
   forall (x1:Ti_1) (x2:Ti_2) ... (xni:Ti_ni),         (induction arguments)
   (HI: I prm1..prmp x1...xni)                         (optional main induction arg)
   -> (Qi x1...xni HI        (f prm1...prmp x1...xni)).(conclusion)
                   ^^        ^^^^^^^^^^^^^^^^^^^^^^^^
               optional                optional
               even if HI      argument added if principle
             present above   generated by functional induction
             [indarg]          [farg]

  HI is not present when the induction principle does not come directly from an
  inductive type (like when it is generated by functional induction for
  example). HI is present otherwise BUT may not appear in the conclusion
  (dependent principle). HI and (f...) cannot be both present.

  Principles taken from functional induction have the final (f...).
*)

(* [rel_contexts] and [rel_declaration] actually contain triples, and
   lists are actually in reverse order to fit [compose_prod]. *)
type elim_scheme = { 
  elimc: Term.constr * constr Rawterm.bindings;
  elimt: types;
  indref: global_reference option;
  params: rel_context;     (* (prm1,tprm1);(prm2,tprm2)...(prmp,tprmp) *)
  nparams: int;            (* number of parameters *)
  predicates: rel_context; (* (Qq, (Tq_1 -> Tq_2 ->...-> Tq_nq)), (Q1,...) *)
  branches: rel_context;    (* branchr,...,branch1 *)
  args: rel_context;       (* (xni, Ti_ni) ... (x1, Ti_1) *)
  nargs: int;              (* number of arguments *)
  indarg: rel_declaration option; (* Some (H,I prm1..prmp x1...xni) 
				     if HI is in premisses, None otherwise *)
  concl: types;            (* Qi x1...xni HI (f...), HI and (f...) 
			      are optional and mutually exclusive *)
  indarg_in_concl: bool;   (* true if HI appears at the end of conclusion *)
  farg_in_concl: bool;     (* true if (f...) appears at the end of conclusion *)
}

let empty_scheme = 
  { 
    elimc = mkProp,NoBindings;
    elimt = mkProp;
    indref = None;
    params = [];
    nparams = 0;
    predicates = [];
    branches = [];
    args = [];
    nargs = 0;
    indarg = None;
    concl = mkProp;
    indarg_in_concl = false;
    farg_in_concl = false;
  }

exception NoLastArg
exception NoLastArgCcl

(* Builds an elim_scheme frome its type and calling form (const+binding) We
   first separate branches.  We obtain branches, hyps before (params + preds),
   hyps after (args <+ indarg if present>) and conclusion.  Then we proceed as
   follows:
   
   - separate parameters and predicates in params_preds. For that we build: 
 forall (x1:Ti_1)(xni:Ti_ni) (HI:I prm1..prmp x1...xni), DUMMY x1...xni HI/farg
                             ^^^^^^^^^^^^^^^^^^^^^^^^^                  ^^^^^^^
                                       optional                           opt
     Free rels appearing in this term are parameters (branches should not
     appear, and the only predicate would have been Qi but we replaced it by
     DUMMY). We guess this heuristic catches all params.  TODO: generalize to
     the case where args are merged with branches (?) and/or where several
     predicates are cited in the conclusion.

   - finish to fill in the elim_scheme: indarg/farg/args and finally indref. *)
let compute_elim_sig elimc elimt =
  let params_preds,branches,args_indargs,conclusion = 
    decompose_paramspred_branch_args elimt in
  
  let ccl = exchange_hd_app (mkVar (id_of_string "__QI_DUMMY__")) conclusion in
  let concl_with_args = it_mkProd_or_LetIn ccl args_indargs in  
  let nparams = Intset.cardinal (free_rels concl_with_args) in
  let preds,params = cut_list (List.length params_preds - nparams) params_preds in

  (* A first approximation, further anlysis will tweak it *)
  let res = ref { empty_scheme with
    (* This fields are ok: *)
    elimc = elimc; elimt = elimt; concl = conclusion;
    predicates = preds; branches = branches; 
    farg_in_concl = (try isApp (last_arg ccl) with _ -> false);
    params = params; nparams = nparams; 
    (* This fields (+ indarg + indref + indarg_in_concl + farg_in_concl)
       are approximate at this point: *)
    args = args_indargs; nargs = List.length args_indargs; } in
  try 
    (* Order of following tests is important. Each of them exits if successful. *)
    (* 1- First see if (f x...) is in the conclusion. *)
    if !res.farg_in_concl 
    then begin
      res := { !res with
	indarg = None;
	indarg_in_concl = false; farg_in_concl = true };
      raise Exit
    end;
    (* 2- If no args_indargs then no indarg (indarg cannot be a parameter). *)
    if !res.nargs=0 then raise Exit;
    (* 3- Look at last arg: is it the indarg? *)
    ignore (
      match List.hd args_indargs with
	| hiname,Some _,hi -> error "cannot recognize an induction schema"
	| hiname,None,hi -> 
	    let hi_ind, hi_args = decompose_app hi in
	    let hi_is_ind = (* hi est d'un type inductif *)
	      match kind_of_term hi_ind with | Ind (mind,_)  -> true | _ -> false in
	    let hi_args_enough = (* hi a le bon nbre d'arguments *)
	      List.length hi_args = List.length params + !res.nargs -1 in
	    (* FIXME: Ces deux tests ne sont pas suffisants. *)
	    if not (hi_is_ind & hi_args_enough)
	    then raise Exit (* No indarg, args and params unchanged *)
	    else (* Last arg is the indarg *)
	      res := {!res with
		indarg = Some (List.hd !res.args);
		indarg_in_concl = occur_rel 1 ccl;
		args = List.tl !res.args;
		nargs = !res.nargs - 1;
	      };
	    raise Exit);
    raise Exit
  with Exit -> (* Ending by computing indrev: *)
    match !res.indarg with
      | None -> !res (* No indref *)
      | Some ( _,Some _,_) -> error "Cannot recognise an induction scheme"
      | Some ( _,None,ind) -> 
	  let indhd,indargs = decompose_app ind in
	  try {!res with indref = Some (global_of_constr indhd) }
	  with _ -> error "Cannot find the inductive type of the inductive schema";;

(* Check that the elimination scheme has a form similar to the 
   elimination schemes built by Coq. Schemes may have the standard
   form computed from an inductive type OR (feb. 2006) a non standard
   form. That is: with no main induction argument and with an optional
   extra final argument of the form (f x y ...) in the conclusion. In
   the non standard case, naming of generated hypos is slightly
   different. *)
let compute_elim_signature elimc elimt names_info =
  let scheme = compute_elim_sig elimc elimt in
  let f,l = decompose_app scheme.concl in
  (* Vérifier que les arguments de Qi sont bien les xi. *)
  match scheme.indarg with
    | Some (_,Some _,_) -> error "strange letin, cannot recognize an induction schema"
    | None -> (* Non standard scheme *)
	let npred = List.length scheme.predicates in 
	let is_pred n c = 
	  let hd = fst (decompose_app c) in match kind_of_term hd with
	    | Rel q when n < q & q <= n+npred -> IndArg
	    | _ -> OtherArg in 
	let rec check_branch p c = 
	  match kind_of_term c with
	    | Prod (_,t,c) -> is_pred p t :: check_branch (p+1) c
	    | LetIn (_,_,_,c) -> OtherArg :: check_branch (p+1) c
	    | _ when is_pred p c = IndArg -> []
	    | _ -> raise Exit in 
	let rec find_branches p lbrch = 
	  match lbrch with
	    | (_,None,t)::brs ->
		(try
		  let lchck_brch = check_branch p t in
		  let n = List.fold_left 
		    (fun n b -> if b=RecArg then n+1 else n) 0 lchck_brch in
		  let recvarname, hyprecname, avoid = 
		    make_up_names n scheme.indref names_info in
		  let namesign = 
		    List.map (fun b -> (b,if b=IndArg then hyprecname else recvarname))
		      lchck_brch in
		  (avoid,namesign) :: find_branches (p+1) brs
		with Exit-> error_ind_scheme "the branches of")
	    | (_,Some _,_)::_ -> error_ind_scheme "the branches of"
	    | [] -> [] in
	let indsign = Array.of_list (find_branches 0 (List.rev scheme.branches)) in
	indsign,scheme	
	
    | Some ( _,None,ind) -> (* Standard scheme from an inductive type *)
	let indhd,indargs = decompose_app ind in
	let npred = List.length scheme.predicates in
	let is_pred n c = 
	  let hd = fst (decompose_app c) in match kind_of_term hd with
	    | Rel q when n < q & q <= n+npred -> IndArg
	    | _ when hd = indhd -> RecArg
	    | _ -> OtherArg in
	let rec check_branch p c = match kind_of_term c with
	  | Prod (_,t,c) -> is_pred p t :: check_branch (p+1) c
	  | LetIn (_,_,_,c) -> OtherArg :: check_branch (p+1) c
	  | _ when is_pred p c = IndArg -> []
	  | _ -> raise Exit in 
	let rec find_branches p lbrch =
	  match lbrch with
	    | (_,None,t)::brs ->
		(try
		  let lchck_brch = check_branch p t in
		  let n = List.fold_left 
		    (fun n b -> if b=RecArg then n+1 else n) 0 lchck_brch in
		  let recvarname, hyprecname, avoid = 
		    make_up_names n scheme.indref names_info in
		  let namesign = 
		    List.map (fun b -> (b,if b=IndArg then hyprecname else recvarname))
		      lchck_brch in
		  (avoid,namesign) :: find_branches (p+1) brs
		with Exit -> error_ind_scheme "the branches of")
	    | (_,Some _,_)::_ -> error_ind_scheme "the branches of"
	    | [] ->
		(* Check again conclusion *)

		let ccl_arg_ok = is_pred (p + scheme.nargs + 1) f = IndArg in
		let ind_is_ok = 
		  list_lastn scheme.nargs indargs 
		  = extended_rel_list 0 scheme.args in
		if not (ccl_arg_ok & ind_is_ok) then
		  error "Cannot recognize the conclusion of an induction schema";
		[] 
	in
	let indsign = Array.of_list (find_branches 0 (List.rev scheme.branches)) in
	indsign,scheme


let find_elim_signature isrec elim hyp0 gl =
  let tmptyp0 =	pf_get_hyp_typ gl hyp0 in
  let (elimc,elimt) = match elim with
    | None ->
	let mind,_ = pf_reduce_to_quantified_ind gl tmptyp0 in
	let s = elimination_sort_of_goal gl in
	let elimc =
	  if isrec then lookup_eliminator mind s
	  else pf_apply make_case_gen gl mind s in
	let elimt = pf_type_of gl elimc in
	((elimc, NoBindings), elimt)
    | Some (elimc,lbind as e) -> 
	(e, pf_type_of gl elimc) in
  let indsign,elim_scheme = compute_elim_signature elimc elimt hyp0 in
  (indsign,elim_scheme)


let induction_from_context isrec elim_info hyp0 names gl =
  (*test suivant sans doute inutile car refait par le letin_tac*)
  if List.mem hyp0 (ids_of_named_context (Global.named_context())) then
    errorlabstrm "induction" 
      (str "Cannot generalize a global variable");
  let indsign,scheme = elim_info in

  let indref = match scheme.indref with | None -> assert false | Some x -> x in
  let tmptyp0 =	pf_get_hyp_typ gl hyp0 in
  let typ0 = pf_apply reduce_to_quantified_ref gl indref tmptyp0 in

  let env = pf_env gl in
  let indvars = find_atomic_param_of_ind scheme.nparams (snd (decompose_prod typ0)) in
  let statlists,lhyp0,indhyps,deps = cook_sign (Some hyp0) indvars env in
  let tmpcl = it_mkNamedProd_or_LetIn (pf_concl gl) deps in
  let names = compute_induction_names (Array.length indsign) names in
  let dephyps = List.map (fun (id,_,_) -> id) deps in
  let deps_cstr =
    List.fold_left
      (fun a (id,b,_) -> if b = None then (mkVar id)::a else a) [] deps in

  (* Magistral effet de bord: si hyp0 a des arguments, ceux d'entre
     eux qui ouvrent de nouveaux buts arrivent en premier dans la
     liste des sous-buts du fait qu'ils sont le plus à gauche dans le
     combinateur engendré par make_case_gen (un "Cases (hyp0 ?) of
     ...")  et il faut alors appliquer tclTHENLASTn; en revanche,
     comme lookup_eliminator renvoie un combinateur de la forme
     "ind_rec ... (hyp0 ?)", les buts correspondant à des arguments de
     hyp0 sont maintenant à la fin et c'est tclTHENFIRSTn qui marche !!! *)
  tclTHENLIST
    [ if deps = [] then tclIDTAC else apply_type tmpcl deps_cstr;
      thin dephyps;
      (if isrec then tclTHENFIRSTn else tclTHENLASTn)
       	(tclTHENLIST
	  [ induction_tac hyp0 typ0 (scheme.elimc,scheme.elimt);
	    thin [hyp0];
            tclTRY (thin indhyps) ])
       	(array_map2
	   (induct_discharge statlists lhyp0 (List.rev dephyps)) indsign names)
    ]
    gl


let mapi f l =
  let rec mapi_aux f i l =   
    match l with
      | [] -> []
      | e::l' -> f e i :: mapi_aux f (i+1) l' in
  mapi_aux f 0 l


(* Instanciate all meta variables of elimclause using lid, some elts
   of lid are parameters (first ones), the other are
   arguments. Returns the clause obtained.  *)
let recolle_clenv scheme lid elimclause gl = 
  let _,arr = destApp elimclause.templval.rebus in
  let lindmv = 
    Array.map
      (fun x -> 
	match kind_of_term x with
	  | Meta mv -> mv
	  | _  -> errorlabstrm "elimination_clause"
              (str "The type of elimination clause is not well-formed"))
      arr in
  let nmv = Array.length lindmv in
  let lidparams,lidargs = cut_list (scheme.nparams) lid in
  (* parameters correspond to first elts of lid. *)
  let clauses_params = 
    mapi (fun id i -> mkVar id , pf_get_hyp_typ gl id , lindmv.(i)) lidparams in
  (* arguments correspond to last elts of lid. *)
  let clauses_args = 
    mapi 
      (fun id i -> mkVar id , pf_get_hyp_typ gl id , lindmv.(nmv-i-1))
      lidargs in
  let clause_indarg = 
    match scheme.indarg with
      | None -> []
      | Some (x,_,typx) -> []
  in
  let clauses = clauses_params@clauses_args@clause_indarg in
  (* iteration of clenv_fchain with all infos we have. *)
  List.fold_right
    (fun e acc ->
      let x,y,i = e in
      let indclause  = make_clenv_binding gl (x,y) NoBindings in
      let elimclause' = clenv_fchain i acc indclause in
      elimclause')
    (List.rev clauses)
    elimclause

(* Unification of the goal and the principle applied to meta variables:
   (elimc ?i ?j ?k...?l). This solves partly meta variables (and may
    produce new ones). Then refine with the resulting term with holes.
*)
let induction_tac_felim indvars (elimc,lbindelimc) elimt scheme gl = 
  (* elimclause contains this: (elimc ?i ?j ?k...?l) *)
  let elimclause =
    make_clenv_binding gl (mkCast (elimc,DEFAULTcast, elimt),elimt) lbindelimc in
  (* elimclause' is built from elimclause by instanciating all args and params. *)
  let elimclause' = recolle_clenv scheme indvars elimclause gl in
  (* one last resolution (useless?) *)
  let resolved = clenv_unique_resolver true elimclause' gl in
  clenv_refine resolved gl

(* Induction with several induction arguments, main differences with
   induction_from_context is that there is no main induction argument,
   so we chose one to be the positioning reference. On the other hand,
   all args and params must be given, so we help a bit the unifier by
   making the "pattern" by hand before calling induction_tac_felim
   FIXME: REUNIF AVEC induction_tac_felim? *)
let induction_from_context_l isrec elim_info lid names gl =
  let indsign,scheme = elim_info in
  (* hyp0 is the first element of the list of variables on which to
     induct. It is most probably the first of them appearing in the
     context. So it seems to be a good value for hyp0, which is used
     for re-introducing hyps at the right place afterward. *)
  let env = pf_env gl in
  let hyp0,indvars,lid_params = 
    match lid with
      | []  -> anomaly "induction_from_context_l"
      | e::l -> 
	  let nargs_without_indarg = 
	    scheme.nargs - 1
	    + (if scheme.farg_in_concl then 1 else 0) 
	    + (if scheme.indarg <> None then 1 else 0) in
	  let ivs,lp = cut_list nargs_without_indarg l in
	  e, ivs, lp in
  let statlists,lhyp0,indhyps,deps = cook_sign None (hyp0::indvars) env in
  let tmpcl = it_mkNamedProd_or_LetIn (pf_concl gl) deps in
  let names = compute_induction_names (Array.length indsign) names in
  let dephyps = List.map (fun (id,_,_) -> id) deps in
  let deps_cstr =
    List.fold_left
      (fun a (id,b,_) -> if b = None then (mkVar id)::a else a) [] deps in
  (* terms to patternify we must patternify indarg or farg if present in concl *)
  let lid_in_pattern = 
    if scheme.indarg <> None & not scheme.indarg_in_concl then indvars
    else hyp0::indvars in
  let lidcstr = List.map (fun x -> mkVar x) lid_in_pattern in
  let realindvars = (* hyp0 is a real induction arg if it is not the
		       farg in the conclusion of the induction scheme *)
    lid_params @ (if scheme.farg_in_concl then indvars else hyp0::indvars) in
  (* Magistral effet de bord: comme dans induction_from_context. *)
  tclTHENLIST
    [ 
      (* Generalize dependent hyps (but not args) *)
      if deps = [] then tclIDTAC else apply_type tmpcl deps_cstr;
      thin dephyps; (* clear dependent hyps *)
      (* pattern to make the predicate appear. *)
      reduce (Pattern (List.map (fun e -> ([],e)) (List.rev lidcstr))) onConcl;
      (* FIXME: Tester ca avec un principe dependant et non-dependant *)
      (if isrec then tclTHENFIRSTn else tclTHENLASTn)
       	(tclTHENLIST [ 
	  (* Induction by "refine (indscheme ?i ?j ?k...)" + resolution of all
	     possible holes using arguments given by the user (but the
	     functional one). *)
	  induction_tac_felim realindvars scheme.elimc scheme.elimt scheme;
          tclTRY (thin (List.rev (indhyps)));
	])
	(array_map2 
	  (induct_discharge statlists lhyp0 (List.rev dephyps)) indsign names)
    ]
    gl

exception TryNewInduct

let induction_with_atomization_of_ind_arg isrec elim names hyp0 gl =
  let (indsign,scheme as elim_info) =
    find_elim_signature isrec elim hyp0 gl in
  if scheme.indarg = None then (* This is not a standard induction scheme (the
				  argument is probably a parameter) So try the
				  more general induction mechanism. *)
    induction_from_context_l isrec elim_info [hyp0] names gl
  else
    let indref = match scheme.indref with | None -> assert false | Some x -> x in
    tclTHEN
      (atomize_param_of_ind (indref,scheme.nparams) hyp0)
      (induction_from_context isrec elim_info hyp0 names) gl

(* Induction on a list of induction arguments. Analyse the elim
   scheme (which is mandatory for multiple ind args), check that all
   parameters and arguments are given (mandatory too). *)
let induction_without_atomization isrec elim names lid gl =
  let (indsign,scheme as elim_info) =
    find_elim_signature isrec elim (List.hd lid) gl in
  let awaited_nargs = 
    scheme.nparams + scheme.nargs 
    + (if scheme.farg_in_concl then 1 else 0)
    + (if scheme.indarg <> None then 1 else 0)
  in
  let nlid = List.length lid in
  if nlid <> awaited_nargs
  then error "Not the right number of induction arguments"
  else induction_from_context_l isrec elim_info lid names gl

let new_induct_gen isrec elim names c gl =
  match kind_of_term c with
    | Var id when not (mem_named_context id (Global.named_context())) ->
	induction_with_atomization_of_ind_arg isrec elim names id gl
    | _        ->
	let x = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c) 
		  Anonymous in
	let id = fresh_id [] x gl in
	tclTHEN
	  (letin_tac true (Name id) c allClauses)
	  (induction_with_atomization_of_ind_arg isrec elim names id) gl

(* Induction on a list of arguments. First make induction arguments
   atomic (using letins), then do induction. The specificity here is
   that all arguments and parameters of the scheme are given
   (mandatory for the moment), so we don't need to deal with
    parameters of the inductive type as in new_induct_gen. *)
let new_induct_gen_l isrec elim names lc gl =
  let newlc = ref [] in
  let rec atomize_list l gl =
    match l with
      | [] -> tclIDTAC gl
      | c::l' ->
	  match kind_of_term c with
	    | Var id when not (mem_named_context id (Global.named_context())) -> 
		let _ = newlc:= id::!newlc in
		atomize_list l' gl
	    | _ ->
		let x = 
		  id_of_name_using_hdchar (Global.env()) (pf_type_of gl c) Anonymous in
		let id = fresh_id [] x gl in
		let _ = newlc:=id::!newlc in
		tclTHEN 
		  (letin_tac true (Name id) c allClauses)
		  (atomize_list l') gl in
  tclTHEN
    (atomize_list lc)
    (fun gl' -> (* recompute each time to have the new value of newlc *)
      induction_without_atomization isrec elim names !newlc gl') gl


(* Induction either over a term, over a quantified premisse, or over
   several quantified premisses (like with functional induction
   principles). *)
let new_induct_destruct_l isrec lc elim names = 
  assert (List.length lc > 0); (* ensured by syntax, but if called inside caml? *)
  try 
    if List.length lc = 1 then (* induction on one arg *)
      try 
	let c = List.hd lc in
	match c with
	  | ElimOnConstr c -> new_induct_gen isrec elim names c
	  | ElimOnAnonHyp n ->
	      tclTHEN (intros_until_n n)
		(tclLAST_HYP (new_induct_gen isrec elim names))
		(* Identifier apart because id can be quantified in goal and not typable *)
	  | ElimOnIdent (_,id) ->
	      tclTHEN (tclTRY (intros_until_id id))
		(new_induct_gen isrec elim names (mkVar id))
      with _ -> raise TryNewInduct
    else raise TryNewInduct
  with TryNewInduct -> 
    (* Several induction hyps: induction scheme is mandatory *)
    let _ = 
      if elim = None 
      then 
	error ("Induction scheme must be given when several induction hypothesis.\n"
	^ "Example: induction x1 x2 x3 using my_scheme.") in
    let newlc = 
      List.map
	(fun x -> 
	  match x with (* FIXME: should we deal with ElimOnIdent? *)
	    | ElimOnConstr x -> x
	    | _ -> error "don't know where to find some argument")
	lc in
    new_induct_gen_l isrec elim names newlc
      

let new_induct = new_induct_destruct_l true
let new_destruct c = new_induct_destruct_l false [c]

(* The registered tactic, which calls the default elimination
 * if no elimination constant is provided. *)
	
(* Induction tactics *)

(* This was Induction before 6.3 (induction only in quantified premisses) *)
let raw_induct s = tclTHEN (intros_until_id s) (tclLAST_HYP simplest_elim)
let raw_induct_nodep n = tclTHEN (intros_until_n n) (tclLAST_HYP simplest_elim)

let simple_induct_id hyp = raw_induct hyp
let simple_induct_nodep = raw_induct_nodep

let simple_induct = function
  | NamedHyp id -> simple_induct_id id
  | AnonHyp n -> simple_induct_nodep n

(* Destruction tactics *)

let simple_destruct_id s    =
  (tclTHEN (intros_until_id s) (tclLAST_HYP simplest_case))
let simple_destruct_nodep n =
  (tclTHEN (intros_until_n n)    (tclLAST_HYP simplest_case))

let simple_destruct = function
  | NamedHyp id -> simple_destruct_id id
  | AnonHyp n -> simple_destruct_nodep n

(*
 *  Eliminations giving the type instead of the proof.
 * These tactics use the default elimination constant and
 * no substitutions at all. 
 * May be they should be integrated into Elim ...
 *)

let elim_scheme_type elim t gl =
  let clause = mk_clenv_type_of gl elim in
  match kind_of_term (last_arg clause.templval.rebus) with
    | Meta mv ->
        let clause' =
	  (* t is inductive, then CUMUL or CONV is irrelevant *)
	  clenv_unify true Reduction.CUMUL t
            (clenv_meta_type clause mv) clause in
	res_pf clause' ~allow_K:true gl
    | _ -> anomaly "elim_scheme_type"

let elim_type t gl =
  let (ind,t) = pf_reduce_to_atomic_ind gl t in
  let elimc = lookup_eliminator ind (elimination_sort_of_goal gl) in
  elim_scheme_type elimc t gl

let case_type t gl =
  let (ind,t) = pf_reduce_to_atomic_ind gl t in
  let env = pf_env gl in
  let elimc = make_case_gen env (project gl) ind (elimination_sort_of_goal gl) in 
  elim_scheme_type elimc t gl


(* Some eliminations frequently used *)

(* These elimination tactics are particularly adapted for sequent
   calculus.  They take a clause as argument, and yield the
   elimination rule if the clause is of the form (Some id) and a
   suitable introduction rule otherwise. They do not depend on 
   the name of the eliminated constant, so they can be also 
   used on ad-hoc disjunctions and conjunctions introduced by
   the user. 
   -- Eduardo Gimenez (11/8/97)

   HH (29/5/99) replaces failures by specific error messages
 *)

let andE id gl =
  let t = pf_get_hyp_typ gl id in
  if is_conjunction (pf_hnf_constr gl t) then 
    (tclTHEN (simplest_elim (mkVar id)) (tclDO 2 intro)) gl
  else 
    errorlabstrm "andE" 
      (str("Tactic andE expects "^(string_of_id id)^" is a conjunction."))

let dAnd cls =
  onClauses
    (function
      | None    -> simplest_split
      | Some (id,_,_) -> andE id)
    cls

let orE id gl =
  let t = pf_get_hyp_typ gl id in
  if is_disjunction (pf_hnf_constr gl t) then 
    (tclTHEN (simplest_elim (mkVar id)) intro) gl
  else 
    errorlabstrm "orE" 
      (str("Tactic orE expects "^(string_of_id id)^" is a disjunction."))

let dorE b cls =
  onClauses
    (function
      | (Some (id,_,_)) -> orE id
      |  None     -> (if b then right else left) NoBindings)
    cls

let impE id gl =
  let t = pf_get_hyp_typ gl id in
  if is_imp_term (pf_hnf_constr gl t) then 
    let (dom, _, rng) = destProd (pf_hnf_constr gl t) in 
    tclTHENLAST
      (cut_intro rng) 
      (apply_term (mkVar id) [mkMeta (new_meta())]) gl
  else 
    errorlabstrm "impE"
      (str("Tactic impE expects "^(string_of_id id)^
	      " is a an implication."))
                        
let dImp cls =
  onClauses
    (function
      | None    -> intro
      | Some (id,_,_) -> impE id)
    cls

(************************************************)
(*  Tactics related with logic connectives      *)
(************************************************)

(* Reflexivity tactics *)

let setoid_reflexivity = ref (fun _ -> assert false)
let register_setoid_reflexivity f = setoid_reflexivity := f

let reflexivity gl =
  match match_with_equation (pf_concl gl) with
    | None -> !setoid_reflexivity gl
    | Some (hdcncl,args) ->  one_constructor 1 NoBindings gl

let intros_reflexivity  = (tclTHEN intros reflexivity)

(* Symmetry tactics *)

(* This tactic first tries to apply a constant named sym_eq, where eq
   is the name of the equality predicate. If this constant is not
   defined and the conclusion is a=b, it solves the goal doing (Cut
   b=a;Intro H;Case H;Constructor 1) *)

let setoid_symmetry = ref (fun _ -> assert false)
let register_setoid_symmetry f = setoid_symmetry := f

let symmetry gl =
  match match_with_equation (pf_concl gl) with
    | None -> !setoid_symmetry gl
    | Some (hdcncl,args) ->
        let hdcncls = string_of_inductive hdcncl in
        begin 
	  try 
	    (apply (pf_parse_const gl ("sym_"^hdcncls)) gl)
          with  _ ->
            let symc = match args with 
	      | [t1; c1; t2; c2] -> mkApp (hdcncl, [| t2; c2; t1; c1 |])
              | [typ;c1;c2] -> mkApp (hdcncl, [| typ; c2; c1 |])
              | [c1;c2]     -> mkApp (hdcncl, [| c2; c1 |])
	      | _ -> assert false 
	    in 
	    tclTHENLAST (cut symc)
              (tclTHENLIST 
		[ intro;
		  tclLAST_HYP simplest_case;
		  one_constructor 1 NoBindings ])
	      gl
	end

let setoid_symmetry_in = ref (fun _ _ -> assert false)
let register_setoid_symmetry_in f = setoid_symmetry_in := f

let symmetry_in id gl = 
  let ctype = pf_type_of gl (mkVar id) in 
  let sign,t = decompose_prod_assum ctype in
  match match_with_equation t with
    | None -> !setoid_symmetry_in id gl
    | Some (hdcncl,args) -> 
        let symccl = match args with 
	  | [t1; c1; t2; c2] -> mkApp (hdcncl, [| t2; c2; t1; c1 |])
          | [typ;c1;c2] -> mkApp (hdcncl, [| typ; c2; c1 |])
          | [c1;c2]     -> mkApp (hdcncl, [| c2; c1 |])
	  | _ -> assert false in
	tclTHENS (cut (it_mkProd_or_LetIn symccl sign))
	  [ intro_replacing id;
            tclTHENLIST [ intros; symmetry; apply (mkVar id); assumption ] ]
	  gl

let intros_symmetry =
  onClauses
    (function
      | None -> tclTHEN intros symmetry
      | Some (id,_,_) -> symmetry_in id)

(* Transitivity tactics *)

(* This tactic first tries to apply a constant named trans_eq, where eq
   is the name of the equality predicate. If this constant is not
   defined and the conclusion is a=b, it solves the goal doing 
   Cut x1=x2; 
       [Cut x2=x3; [Intros e1 e2; Case e2;Assumption 
                    | Idtac]
       | Idtac]
   --Eduardo (19/8/97)
*)

let setoid_transitivity = ref (fun _ _ -> assert false)
let register_setoid_transitivity f = setoid_transitivity := f

let transitivity t gl =
  match match_with_equation (pf_concl gl) with
    | None -> !setoid_transitivity t gl
    | Some (hdcncl,args) -> 
        let hdcncls = string_of_inductive hdcncl in
        begin
	  try 
	    apply_list [(pf_parse_const gl ("trans_"^hdcncls));t] gl 
          with  _ -> 
            let eq1, eq2 = match args with 
	      | [typ1;c1;typ2;c2] -> let typt = pf_type_of gl t in
                  ( mkApp(hdcncl, [| typ1; c1; typt ;t |]),
		    mkApp(hdcncl, [| typt; t; typ2; c2 |]) )
              | [typ;c1;c2] ->
		  ( mkApp (hdcncl, [| typ; c1; t |]),
		    mkApp (hdcncl, [| typ; t; c2 |]) )
	      | [c1;c2]     ->
		  ( mkApp (hdcncl, [| c1; t|]),
		    mkApp (hdcncl, [| t; c2 |]) )
	      | _ -> assert false 
	    in
            tclTHENFIRST (cut eq2)
	      (tclTHENFIRST (cut eq1)
                (tclTHENLIST
		  [ tclDO 2 intro;
		    tclLAST_HYP simplest_case;
		    assumption ])) gl
        end 
	
let intros_transitivity  n  = tclTHEN intros (transitivity n)

(* tactical to save as name a subproof such that the generalisation of 
   the current goal, abstracted with respect to the local signature, 
   is solved by tac *)

let interpretable_as_section_decl d1 d2 = match d1,d2 with
  | (_,Some _,_), (_,None,_) -> false
  | (_,Some b1,t1), (_,Some b2,t2) -> eq_constr b1 b2 & eq_constr t1 t2
  | (_,None,t1), (_,_,t2) -> eq_constr t1 t2

let abstract_subproof name tac gls = 
  let current_sign = Global.named_context()
  and global_sign = pf_hyps gls in
  let sign,secsign = 
    List.fold_right
      (fun (id,_,_ as d) (s1,s2) -> 
	 if mem_named_context id current_sign &
           interpretable_as_section_decl (Sign.lookup_named id current_sign) d
         then (s1,push_named_context_val d s2)
	 else (add_named_decl d s1,s2)) 
      global_sign (empty_named_context,empty_named_context_val) in
  let na = next_global_ident_away false name (pf_ids_of_hyps gls) in
  let concl = it_mkNamedProd_or_LetIn (pf_concl gls) sign in
  if occur_existential concl then
    error "\"abstract\" cannot handle existentials";
  let lemme =
    start_proof na (Global, Proof Lemma) secsign concl (fun _ _ -> ());
    let _,(const,kind,_) =
      try
	by (tclCOMPLETE (tclTHEN (tclDO (List.length sign) intro) tac)); 
	let r = cook_proof () in 
	delete_current_proof (); r
      with e when catchable_exception e -> 
	(delete_current_proof(); raise e)
    in   (* Faudrait un peu fonctionnaliser cela *)
    let cd = Entries.DefinitionEntry const in
    let con = Declare.declare_internal_constant na (cd,IsProof Lemma) in
    constr_of_global (ConstRef con)
  in
  exact_no_check 
    (applist (lemme, 
              List.rev (Array.to_list (instance_from_named_context sign))))
    gls

let tclABSTRACT name_op tac gls = 
  let s = match name_op with 
    | Some s -> s 
    | None   -> add_suffix (get_current_proof_name ()) "_subproof" 
  in  
  abstract_subproof s tac gls


let admit_as_an_axiom gls =
  let current_sign = Global.named_context()
  and global_sign = pf_hyps gls in
  let sign,secsign = 
    List.fold_right
      (fun (id,_,_ as d) (s1,s2) -> 
	 if mem_named_context id current_sign &
           interpretable_as_section_decl (Sign.lookup_named id current_sign) d
         then (s1,add_named_decl d s2)
	 else (add_named_decl d s1,s2)) 
      global_sign (empty_named_context,empty_named_context) in
  let name = add_suffix (get_current_proof_name ()) "_admitted" in
  let na = next_global_ident_away false name (pf_ids_of_hyps gls) in
  let concl = it_mkNamedProd_or_LetIn (pf_concl gls) sign in
  if occur_existential concl then error "\"admit\" cannot handle existentials";
  let axiom =
    let cd = Entries.ParameterEntry concl in
    let con = Declare.declare_internal_constant na (cd,IsAssumption Logical) in
    constr_of_global (ConstRef con)
  in
  exact_no_check 
    (applist (axiom, 
              List.rev (Array.to_list (instance_from_named_context sign))))
    gls