open Context.Rel.Declaration
open CErrors
open Util
open Names
open Term
open Pp
open Indfun_common
open Libnames
open Globnames
open Glob_term
open Declarations
open Misctypes
open Decl_kinds
open Sigma.Notations

let is_rec_info scheme_info =
  let test_branche min acc decl =
    acc || (
      let new_branche =
	it_mkProd_or_LetIn mkProp (fst (decompose_prod_assum (get_type decl))) in
      let free_rels_in_br = Termops.free_rels new_branche in
      let max = min + scheme_info.Tactics.npredicates in
      Int.Set.exists (fun i -> i >= min && i< max) free_rels_in_br
    )
  in
  List.fold_left_i test_branche 1 false (List.rev scheme_info.Tactics.branches)

let choose_dest_or_ind scheme_info =
  Tactics.induction_destruct (is_rec_info scheme_info) false

let functional_induction with_clean c princl pat =
  let res =
    let f,args = decompose_app c in
    fun g ->
    let princ,bindings, princ_type,g' =
      match princl with
  No principle is given let 's find the good one
	 begin
	   match kind_of_term f with
	   | Const (c',u) ->
	      let princ_option =
  we first try to find out a graph on f
		  try find_Function_infos c'
		  with Not_found ->
		    errorlabstrm "" (str "Cannot find induction information on "++
				       Printer.pr_lconstr (mkConst c') )
		in
		match Tacticals.elimination_sort_of_goal g with
		| InProp -> finfo.prop_lemma
		| InSet -> finfo.rec_lemma
		| InType -> finfo.rect_lemma
	      in
	      let princ,g' =  (* then we get the principle *)
		try
		  let g',princ =
		    Tacmach.pf_eapply (Evd.fresh_global) g  (Globnames.ConstRef (Option.get princ_option )) in
		  princ,g'
		with Option.IsNone ->
		  (*i If there is not default lemma defined then,
			  we cross our finger and try to find a lemma named f_ind
			  (or f_rec, f_rect) i*)
		  let princ_name =
		    Indrec.make_elimination_ident
		      (Label.to_id (con_label c'))
		      (Tacticals.elimination_sort_of_goal g)
		  in
		  try
		    let princ_ref = const_of_id princ_name in
		    let (a,b) = Tacmach.pf_eapply (Evd.fresh_global) g princ_ref in
		    (b,a)
  mkConst(const_of_id princ_name ) ,  ( \ * FIXME * \ )
		  with Not_found -> (* This one is neither defined ! *)
		    errorlabstrm "" (str "Cannot find induction principle for "
				     ++Printer.pr_lconstr (mkConst c') )
	      in
	      (princ,NoBindings, Tacmach.pf_unsafe_type_of g' princ,g')
	   | _ -> raise (UserError("",str "functional induction must be used with a function" ))
	 end
      | Some ((princ,binding)) ->
	 princ,binding,Tacmach.pf_unsafe_type_of g princ,g
    in
    let princ_infos = Tactics.compute_elim_sig princ_type in
    let args_as_induction_constr =
      let c_list =
	if princ_infos.Tactics.farg_in_concl
	then [c] else []
      in
      let encoded_pat_as_patlist =
        List.make (List.length args + List.length c_list - 1) None @ [pat] in
      List.map2 (fun c pat -> ((None,Tacexpr.ElimOnConstr ({ Tacexpr.delayed = fun env sigma -> Sigma ((c,NoBindings), sigma, Sigma.refl) })),(None,pat),None))
        (args@c_list) encoded_pat_as_patlist
    in
    let princ' = Some (princ,bindings) in
    let princ_vars =
      List.fold_right
	(fun a acc -> try Id.Set.add (destVar a) acc with DestKO -> acc)
	args
	Id.Set.empty
    in
    let old_idl = List.fold_right Id.Set.add (Tacmach.pf_ids_of_hyps g) Id.Set.empty in
    let old_idl = Id.Set.diff old_idl princ_vars in
    let subst_and_reduce g =
      if with_clean
      then
	let idl =
	  List.filter (fun id -> not (Id.Set.mem id old_idl))
	    (Tacmach.pf_ids_of_hyps g)
	in
	let flag =
	  Genredexpr.Cbv
	    {Redops.all_flags
	     with Genredexpr.rDelta = false;
	    }
	in
	Tacticals.tclTHEN
	  (Tacticals.tclMAP (fun id -> Tacticals.tclTRY (Proofview.V82.of_tactic (Equality.subst_gen (do_rewrite_dependent ()) [id]))) idl )
	  (Proofview.V82.of_tactic (Tactics.reduce flag Locusops.allHypsAndConcl))
	  g
      else Tacticals.tclIDTAC g
    in
    Tacticals.tclTHEN
      (Proofview.V82.of_tactic (choose_dest_or_ind
	 princ_infos
	 (args_as_induction_constr,princ')))
      subst_and_reduce
      g'
  in res

let rec abstract_glob_constr c = function
  | [] -> c
  | Constrexpr.LocalRawDef (x,b)::bl -> Constrexpr_ops.mkLetInC(x,b,abstract_glob_constr c bl)
  | Constrexpr.LocalRawAssum (idl,k,t)::bl ->
      List.fold_right (fun x b -> Constrexpr_ops.mkLambdaC([x],k,t,b)) idl
        (abstract_glob_constr c bl)
  | Constrexpr.LocalPattern _::bl -> assert false

let interp_casted_constr_with_implicits env sigma impls c  =
  Constrintern.intern_gen Pretyping.WithoutTypeConstraint env ~impls
    ~allow_patvar:false c


    Construct a fixpoint as a Glob_term 
    and not as a constr 

   Construct a fixpoint as a Glob_term
   and not as a constr
*)

let build_newrecursive
    lnameargsardef  =
  let env0 = Global.env() in
  let sigma = Evd.from_env env0 in
  let (rec_sign,rec_impls) =
    List.fold_left
      (fun (env,impls) (((_,recname),_),bl,arityc,_) ->
        let arityc = Constrexpr_ops.prod_constr_expr arityc bl in
        let arity,ctx = Constrintern.interp_type env0 sigma arityc in
	let evdref = ref (Evd.from_env env0) in
	let _, (_, impls') = Constrintern.interp_context_evars env evdref bl in
	let impl = Constrintern.compute_internalization_data env0 Constrintern.Recursive arity impls' in
        let open Context.Named.Declaration in
        (Environ.push_named (LocalAssum (recname,arity)) env, Id.Map.add recname impl impls))
      (env0,Constrintern.empty_internalization_env) lnameargsardef in
  let recdef =
    (* Declare local notations *)
    let f (_,bl,_,def) =
      let def = abstract_glob_constr def bl in
      interp_casted_constr_with_implicits
        rec_sign sigma rec_impls def
    in
    States.with_state_protection (List.map f) lnameargsardef
  in
  recdef,rec_impls

let build_newrecursive l = 
  let l' = List.map 
    (fun ((fixna,_,bll,ar,body_opt),lnot) -> 
       match body_opt with 
	 | Some body -> 
	     (fixna,bll,ar,body)
	 | None -> user_err_loc (Loc.ghost,"Function",str "Body of Function must be given")
    ) l
  in
  build_newrecursive l'

(* Checks whether or not the mutual bloc is recursive *)
let is_rec names =
  let names = List.fold_right Id.Set.add names Id.Set.empty in
  let check_id id names =  Id.Set.mem id names in
  let rec lookup names = function
    | GVar(_,id) -> check_id id names
    | GRef _ | GEvar _ | GPatVar _ | GSort _ |  GHole _ -> false
    | GCast(_,b,_) -> lookup names b
    | GRec _ -> error "GRec not handled"
    | GIf(_,b,_,lhs,rhs) ->
	(lookup names b) || (lookup names lhs) || (lookup names rhs)
    | GLetIn(_,na,t,b) | GLambda(_,na,_,t,b) | GProd(_,na,_,t,b)  ->
	lookup names t || lookup (Nameops.name_fold Id.Set.remove na names) b
    | GLetTuple(_,nal,_,t,b) -> lookup names t ||
	lookup
	  (List.fold_left
	     (fun acc na -> Nameops.name_fold Id.Set.remove na acc)
	     names
	     nal
	  )
	  b
    | GApp(_,f,args) -> List.exists (lookup names) (f::args)
    | GCases(_,_,_,el,brl) ->
	List.exists (fun (e,_) -> lookup names e) el ||
	  List.exists (lookup_br names) brl
  and lookup_br names (_,idl,_,rt) =
    let new_names = List.fold_right Id.Set.remove idl names in
    lookup new_names rt
  in
  lookup names

let rec local_binders_length = function
  (* Assume that no `{ ... } contexts occur *)
  | [] -> 0
  | Constrexpr.LocalRawDef _::bl -> 1 + local_binders_length bl
  | Constrexpr.LocalRawAssum (idl,_,_)::bl -> List.length idl + local_binders_length bl
  | Constrexpr.LocalPattern _::bl -> assert false

let prepare_body ((name,_,args,types,_),_) rt =
  let n = local_binders_length args in
    Pp.msgnl ( str " nb lambda to chop : " + + str ( string_of_int n ) + + fnl ( ) + + Printer.pr_glob_constr rt ) ;
  let fun_args,rt' = chop_rlambda_n n rt in
  (fun_args,rt')

let process_vernac_interp_error e =
  fst (ExplainErr.process_vernac_interp_error (e, Exninfo.null))

let warn_funind_cannot_build_inversion =
  CWarnings.create ~name:"funind-cannot-build-inversion" ~category:"funind"
    (fun e' -> strbrk "Cannot build inversion information" ++
                 if do_observe () then (fnl() ++ CErrors.print e') else mt ())

let derive_inversion fix_names =
  try
    let evd' = Evd.from_env (Global.env ()) in 
  we first transform the fix_names identifier into their corresponding constant
    let evd',fix_names_as_constant =
      List.fold_right
	(fun id (evd,l) ->
	 let evd,c =
	   Evd.fresh_global
	     (Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident id)) in 
	 evd, destConst c::l
	)
	fix_names
	(evd',[])
    in

        Then we check that the graphs have been defined 
        If one of the graphs have n't been defined 
        we do nothing 
    
       Then we check that the graphs have been defined
       If one of the graphs haven't been defined
       we do nothing
    *)
    List.iter (fun c -> ignore (find_Function_infos (fst c))) fix_names_as_constant ;
    try
      let evd', lind =
	List.fold_right
	  (fun id (evd,l) ->
	   let evd,id = 
	     Evd.fresh_global
	       (Global.env ()) evd
	       (Constrintern.locate_reference (Libnames.qualid_of_ident (mk_rel_id id)))
	   in 
	   evd,(fst (destInd id))::l
	  )
	  fix_names
	  (evd',[])
      in
      Invfun.derive_correctness
	Functional_principles_types.make_scheme
	functional_induction
	fix_names_as_constant
	lind;
      with e when CErrors.noncritical e ->
      let e' = process_vernac_interp_error e in
      warn_funind_cannot_build_inversion e'
  with e when CErrors.noncritical e ->
    let e' = process_vernac_interp_error e in
    warn_funind_cannot_build_inversion e'

let warn_cannot_define_graph =
  CWarnings.create ~name:"funind-cannot-define-graph" ~category:"funind"
    (fun (names,error) -> strbrk "Cannot define graph(s) for " ++
      h 1 names ++ error)

let warn_cannot_define_principle =
  CWarnings.create ~name:"funind-cannot-define-principle" ~category:"funind"
  (fun (names,error) -> strbrk "Cannot define induction principle(s) for "++
      h 1 names ++ error)

let warning_error names e =
  let e = process_vernac_interp_error e in
  let e_explain e =
    match e with
      | ToShow e -> 
	let e = process_vernac_interp_error e in
	spc () ++ CErrors.print e
      | _ -> 
	if do_observe () 
	then 
	  let e = process_vernac_interp_error e in 
	  (spc () ++ CErrors.print e)
	else mt ()
  in
  match e with
    | Building_graph e ->
       let names = prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names in
       warn_cannot_define_graph (names,e_explain e)
    | Defining_principle e ->
       let names = prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names in
       warn_cannot_define_principle (names,e_explain e)
    | _ -> raise e

let error_error names e =
  let e = process_vernac_interp_error e in
  let e_explain e =
    match e with
      | ToShow e -> spc () ++ CErrors.print e
      | _ -> if do_observe () then (spc () ++ CErrors.print e) else mt ()
  in
  match e with
    | Building_graph e ->
	errorlabstrm ""
	  (str "Cannot define graph(s) for " ++
	     h 1 (prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names) ++
	     e_explain e)
    | _ -> raise e

let generate_principle (evd:Evd.evar_map ref) pconstants on_error
    is_general do_built (fix_rec_l:(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list) list) recdefs  interactive_proof
    (continue_proof : int -> Names.constant array -> Term.constr array -> int ->
      Tacmach.tactic) : unit =
  let names = List.map (function (((_, name),_),_,_,_,_),_ -> name) fix_rec_l in
  let fun_bodies = List.map2 prepare_body fix_rec_l recdefs in
  let funs_args = List.map fst fun_bodies in
  let funs_types =  List.map (function ((_,_,_,types,_),_) -> types) fix_rec_l in
  try
    (* We then register the Inductive graphs of the functions  *)
    Glob_term_to_relation.build_inductive !evd pconstants funs_args funs_types recdefs;
    if do_built
    then
      begin
	(*i The next call to mk_rel_id is valid since we have just construct the graph
	   Ensures by : do_built
	i*)
	let f_R_mut = Ident (Loc.ghost,mk_rel_id (List.nth names 0)) in
	let ind_kn =
	  fst (locate_with_msg
		 (pr_reference f_R_mut++str ": Not an inductive type!")
		 locate_ind
		 f_R_mut)
	in
	let fname_kn (((fname,_),_,_,_,_),_) =
	  let f_ref = Ident fname in
	  locate_with_msg
	    (pr_reference f_ref++str ": Not an inductive type!")
	    locate_constant
	    f_ref
	in
	let funs_kn = Array.of_list (List.map fname_kn fix_rec_l) in
	let _ =
	  List.map_i
	    (fun i x ->
	     let princ = Indrec.lookup_eliminator (ind_kn,i) (InProp) in
	     let env = Global.env () in
	     let evd = ref (Evd.from_env env) in
	     let evd',uprinc = Evd.fresh_global env !evd princ in
             let _ = evd := evd' in 
	     let princ_type = Typing.e_type_of ~refresh:true env evd uprinc in
    	     Functional_principles_types.generate_functional_principle
	       evd
		 interactive_proof
		 princ_type
		 None
		 None
		 (Array.of_list pconstants)
		 (* funs_kn *)
		 i
		 (continue_proof  0 [|funs_kn.(i)|])
	    )
	    0
	    fix_rec_l
	in
	Array.iter (add_Function is_general) funs_kn;
	()
      end
  with e when CErrors.noncritical e ->
    on_error names e

let register_struct is_rec (fixpoint_exprl:(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list) list) =
  match fixpoint_exprl with
    | [(((_,fname),pl),_,bl,ret_type,body),_] when not is_rec ->
      let body = match body with | Some body -> body | None -> user_err_loc (Loc.ghost,"Function",str "Body of Function must be given") in 
      Command.do_definition
	fname
	(Decl_kinds.Global,(Flags.is_universe_polymorphism ()),Decl_kinds.Definition) pl
	bl None body (Some ret_type) (Lemmas.mk_hook (fun _ _ -> ()));
       let evd,rev_pconstants =
	 List.fold_left
	   (fun (evd,l) ((((_,fname),_),_,_,_,_),_) ->
	    let evd,c =
	      Evd.fresh_global
		(Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident fname)) in
	    evd,((destConst c)::l)
	   )
	   (Evd.from_env (Global.env ()),[])
	   fixpoint_exprl
       in
       evd,List.rev rev_pconstants
    | _ ->
       Command.do_fixpoint Global (Flags.is_universe_polymorphism ()) fixpoint_exprl;
       let evd,rev_pconstants =
	 List.fold_left
	   (fun (evd,l) ((((_,fname),_),_,_,_,_),_) ->
	    let evd,c =
	      Evd.fresh_global
		(Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident fname)) in
	    evd,((destConst c)::l)
	   )
	   (Evd.from_env (Global.env ()),[])
	   fixpoint_exprl
       in
       evd,List.rev rev_pconstants
	   

let generate_correction_proof_wf f_ref tcc_lemma_ref
    is_mes functional_ref eq_ref rec_arg_num rec_arg_type nb_args relation
    (_: int) (_:Names.constant array) (_:Term.constr array) (_:int) : Tacmach.tactic =
  Functional_principles_proofs.prove_principle_for_gen
    (f_ref,functional_ref,eq_ref)
    tcc_lemma_ref is_mes  rec_arg_num rec_arg_type relation


let register_wf ?(is_mes=false) fname rec_impls wf_rel_expr wf_arg using_lemmas args ret_type body
    pre_hook
    =
  let type_of_f = Constrexpr_ops.prod_constr_expr ret_type args in
  let rec_arg_num =
    let names =
      List.map
	snd
	(Constrexpr_ops.names_of_local_assums args)
    in
    match wf_arg with
      | None ->
	if Int.equal (List.length names) 1 then 1
	else error "Recursive argument must be specified"
      | Some wf_arg ->
	  List.index Name.equal (Name wf_arg) names
  in
  let unbounded_eq =
    let f_app_args =
      Constrexpr.CAppExpl
	(Loc.ghost,
	 (None,(Ident (Loc.ghost,fname)),None) ,
	 (List.map
	    (function
	       | _,Anonymous -> assert false
	       | _,Name e -> (Constrexpr_ops.mkIdentC e)
	    )
	    (Constrexpr_ops.names_of_local_assums args)
	 )
	)
    in
    Constrexpr.CApp (Loc.ghost,(None,Constrexpr_ops.mkRefC (Qualid (Loc.ghost,(qualid_of_string "Logic.eq")))),
		    [(f_app_args,None);(body,None)])
  in
  let eq = Constrexpr_ops.prod_constr_expr unbounded_eq args in
  let hook ((f_ref,_) as fconst) tcc_lemma_ref (functional_ref,_) (eq_ref,_) rec_arg_num rec_arg_type
      nb_args relation =
    try
      pre_hook [fconst]
	(generate_correction_proof_wf f_ref tcc_lemma_ref is_mes
	   functional_ref eq_ref rec_arg_num rec_arg_type nb_args relation
	);
      derive_inversion [fname]
    with e when CErrors.noncritical e ->
      (* No proof done *)
      ()
  in
  Recdef.recursive_definition
    is_mes fname rec_impls
    type_of_f
    wf_rel_expr
    rec_arg_num
    eq
    hook
    using_lemmas


let register_mes fname rec_impls wf_mes_expr wf_rel_expr_opt wf_arg using_lemmas args ret_type body =
  let wf_arg_type,wf_arg =
    match wf_arg with
      | None ->
	  begin
	    match args with
	      | [Constrexpr.LocalRawAssum ([(_,Name x)],k,t)] -> t,x
	      | _ -> error "Recursive argument must be specified"
	  end
      | Some wf_args ->
	  try
	    match
	      List.find
		(function
		   | Constrexpr.LocalRawAssum(l,k,t) ->
		       List.exists
			 (function (_,Name id) -> Id.equal id wf_args | _ -> false)
			 l
		   | _ -> false
		)
		args
	    with
	      | Constrexpr.LocalRawAssum(_,k,t)  ->	    t,wf_args
	      | _ -> assert false
	  with Not_found -> assert false
  in
  let wf_rel_from_mes,is_mes = 
    match wf_rel_expr_opt with 
      | None ->
	  let ltof =
	    let make_dir l = DirPath.make (List.rev_map Id.of_string l) in
	    Libnames.Qualid (Loc.ghost,Libnames.qualid_of_path
			       (Libnames.make_path (make_dir ["Arith";"Wf_nat"]) (Id.of_string "ltof")))
	  in
	  let fun_from_mes =
	    let applied_mes =
	      Constrexpr_ops.mkAppC(wf_mes_expr,[Constrexpr_ops.mkIdentC wf_arg])    in
	    Constrexpr_ops.mkLambdaC ([(Loc.ghost,Name wf_arg)],Constrexpr_ops.default_binder_kind,wf_arg_type,applied_mes)
	  in
	  let wf_rel_from_mes =
	    Constrexpr_ops.mkAppC(Constrexpr_ops.mkRefC  ltof,[wf_arg_type;fun_from_mes])
	  in
	  wf_rel_from_mes,true
      | Some wf_rel_expr -> 
	  let wf_rel_with_mes = 
	    let a = Names.Id.of_string "___a" in 
	    let b = Names.Id.of_string "___b" in 
	    Constrexpr_ops.mkLambdaC(
	      [Loc.ghost,Name a;Loc.ghost,Name b],
	      Constrexpr.Default Explicit,
	      wf_arg_type,
	      Constrexpr_ops.mkAppC(wf_rel_expr,
			       [
				Constrexpr_ops.mkAppC(wf_mes_expr,[Constrexpr_ops.mkIdentC a]);
				Constrexpr_ops.mkAppC(wf_mes_expr,[Constrexpr_ops.mkIdentC b])
			       ])
			      )
	  in
	  wf_rel_with_mes,false
  in			       
  register_wf ~is_mes:is_mes fname rec_impls wf_rel_from_mes (Some wf_arg)
    using_lemmas args ret_type body

let map_option f = function 
  | None -> None 
  | Some v -> Some (f v)

open Constrexpr
open Topconstr

let make_assoc assoc l1 l2 =
  let fold assoc a b = match a, b with
  | (_, Name na), (_, Name id) -> Id.Map.add na id assoc
  | _, _ -> assoc
  in
  List.fold_left2 fold assoc l1 l2

let rec rebuild_bl (aux,assoc) bl typ = 
	match bl,typ with 
	  | [], _ -> (List.rev aux,replace_vars_constr_expr assoc typ,assoc)
	  | (Constrexpr.LocalRawAssum(nal,bk,_))::bl',typ ->
	     rebuild_nal (aux,assoc) bk bl' nal (List.length nal) typ
	  | (Constrexpr.LocalRawDef(na,_))::bl',Constrexpr.CLetIn(_,_,nat,typ') ->
	    rebuild_bl ((Constrexpr.LocalRawDef(na,replace_vars_constr_expr assoc nat)::aux),assoc)
	      bl' typ'
	  | _ -> assert false
      and rebuild_nal (aux,assoc) bk bl' nal lnal typ = 
	match nal,typ with 
	  | [], _ -> rebuild_bl (aux,assoc) bl' typ
	  | _,CProdN(_,[],typ) -> rebuild_nal (aux,assoc) bk bl' nal lnal typ
	  | _,CProdN(_,(nal',bk',nal't)::rest,typ') -> 
	    let lnal' = List.length nal' in 
	    if lnal' >= lnal 
	    then 
	      let old_nal',new_nal' = List.chop lnal nal' in
	      let nassoc = make_assoc assoc old_nal' nal in
	      let assum = LocalRawAssum(nal,bk,replace_vars_constr_expr assoc nal't) in
	      rebuild_bl ((assum :: aux), nassoc) bl' 
		(if List.is_empty new_nal' && List.is_empty rest
		 then typ'
		 else if List.is_empty new_nal'
		 then CProdN(Loc.ghost,rest,typ')
		 else CProdN(Loc.ghost,((new_nal',bk',nal't)::rest),typ'))
	    else 
	      let captured_nal,non_captured_nal = List.chop lnal' nal in
	      let nassoc = make_assoc assoc nal' captured_nal in
	      let assum = LocalRawAssum(captured_nal,bk,replace_vars_constr_expr assoc nal't) in
	      rebuild_nal ((assum :: aux), nassoc)
		bk bl' non_captured_nal (lnal - lnal') (CProdN(Loc.ghost,rest,typ'))
	  | _ -> assert false

let rebuild_bl (aux,assoc) bl typ = rebuild_bl (aux,assoc) bl typ

let recompute_binder_list (fixpoint_exprl : (Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list) list) = 
  let fixl,ntns = Command.extract_fixpoint_components false fixpoint_exprl in
  let ((_,_,typel),_,ctx,_) = Command.interp_fixpoint fixl ntns in
  let constr_expr_typel = 
    with_full_print (List.map (Constrextern.extern_constr false (Global.env ()) (Evd.from_ctx ctx))) typel in
  let fixpoint_exprl_with_new_bl = 
    List.map2 (fun ((lna,(rec_arg_opt,rec_order),bl,ret_typ,opt_body),notation_list) fix_typ -> 
     
      let new_bl',new_ret_type,_ = rebuild_bl ([],Id.Map.empty) bl fix_typ in 
      (((lna,(rec_arg_opt,rec_order),new_bl',new_ret_type,opt_body),notation_list):(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list))
    )
      fixpoint_exprl constr_expr_typel 
  in 
  fixpoint_exprl_with_new_bl
  

let do_generate_principle pconstants on_error register_built interactive_proof
    (fixpoint_exprl:(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list) list) :unit =
  List.iter (fun (_,l) -> if not (List.is_empty l) then error "Function does not support notations for now") fixpoint_exprl;
  let _is_struct =
    match fixpoint_exprl with
      | [((_,(wf_x,Constrexpr.CWfRec wf_rel),_,_,_),_) as fixpoint_expr] ->
 	  let (((((_,name),pl),_,args,types,body)),_)  as fixpoint_expr =
	    match recompute_binder_list [fixpoint_expr] with 
	      | [e] -> e
	      | _ -> assert false
	  in 
	  let fixpoint_exprl = [fixpoint_expr] in 
	  let body = match body with | Some body -> body | None -> user_err_loc (Loc.ghost,"Function",str "Body of Function must be given") in 
	  let recdefs,rec_impls = build_newrecursive fixpoint_exprl in
	  let using_lemmas = [] in 
	  let pre_hook pconstants =
	    generate_principle
	      (ref (Evd.from_env (Global.env ())))
	      pconstants
	      on_error
	      true
	      register_built
	      fixpoint_exprl
	      recdefs
	      true
	  in
	  if register_built
	  then register_wf name rec_impls wf_rel (map_option snd wf_x) using_lemmas args types body pre_hook;
	  false
      |[((_,(wf_x,Constrexpr.CMeasureRec(wf_mes,wf_rel_opt)),_,_,_),_) as fixpoint_expr] ->
 	  let (((((_,name),_),_,args,types,body)),_)  as fixpoint_expr =
	    match recompute_binder_list [fixpoint_expr] with 
	      | [e] -> e
	      | _ -> assert false
	  in 
	  let fixpoint_exprl = [fixpoint_expr] in 
	  let recdefs,rec_impls = build_newrecursive fixpoint_exprl in
	  let using_lemmas = [] in
	  let body = match body with | Some body -> body | None -> user_err_loc (Loc.ghost,"Function",str "Body of Function must be given") in 
	  let pre_hook pconstants =
	    generate_principle
	      (ref (Evd.from_env (Global.env ())))
	      pconstants
	      on_error
	      true
	      register_built
	      fixpoint_exprl
	      recdefs
	      true
	  in
	  if register_built
	  then register_mes name rec_impls wf_mes wf_rel_opt (map_option snd wf_x) using_lemmas args types body pre_hook;
	  true
      | _ ->
	  List.iter (function ((_na,(_,ord),_args,_body,_type),_not) ->
		       match ord with 
			 | Constrexpr.CMeasureRec _ | Constrexpr.CWfRec _ ->
			     error
			       ("Cannot use mutual definition with well-founded recursion or measure")
			 | _ -> ()
		    )
	    fixpoint_exprl;
	let fixpoint_exprl = recompute_binder_list fixpoint_exprl in 
	let fix_names =
	  List.map (function ((((_,name),_),_,_,_,_),_) -> name) fixpoint_exprl
	in
	(* ok all the expressions are structural *)
  	let recdefs,rec_impls = build_newrecursive fixpoint_exprl in
	let is_rec = List.exists (is_rec fix_names) recdefs in
	let evd,pconstants =
	  if register_built
	  then register_struct is_rec fixpoint_exprl
	  else (Evd.from_env (Global.env ()),pconstants)
	in
	let evd = ref evd in 
	generate_principle
	  (ref !evd)
	  pconstants
	  on_error
	  false
	  register_built
	  fixpoint_exprl
	  recdefs
	  interactive_proof
	  (Functional_principles_proofs.prove_princ_for_struct evd interactive_proof);
	if register_built then begin derive_inversion fix_names; end;
	true; 
  in
  ()

let rec add_args id new_args b =
  match b with
  | CRef (r,_) ->
      begin      match r with
	| Libnames.Ident(loc,fname) when Id.equal fname id ->
	    CAppExpl(Loc.ghost,(None,r,None),new_args)
	| _ -> b
      end
  | CFix  _  | CCoFix _ -> anomaly ~label:"add_args " (Pp.str "todo")
  | CProdN(loc,nal,b1) ->
      CProdN(loc,
	     List.map (fun (nal,k,b2) -> (nal,k,add_args id new_args b2)) nal,
	     add_args id new_args  b1)
  | CLambdaN(loc,nal,b1) ->
      CLambdaN(loc,
	       List.map (fun (nal,k,b2) -> (nal,k,add_args id new_args  b2)) nal,
	       add_args id new_args  b1)
  | CLetIn(loc,na,b1,b2) ->
      CLetIn(loc,na,add_args id new_args b1,add_args id new_args b2)
  | CAppExpl(loc,(pf,r,us),exprl) ->
      begin
	match r with
	| Libnames.Ident(loc,fname) when Id.equal fname id ->
	    CAppExpl(loc,(pf,r,us),new_args@(List.map (add_args id new_args) exprl))
	| _ -> CAppExpl(loc,(pf,r,us),List.map (add_args id new_args) exprl)
      end
  | CApp(loc,(pf,b),bl) ->
      CApp(loc,(pf,add_args id new_args b),
	   List.map (fun (e,o) -> add_args id new_args e,o) bl)
  | CCases(loc,sty,b_option,cel,cal) ->
      CCases(loc,sty,Option.map (add_args id new_args) b_option,
	     List.map (fun (b,na,b_option) ->
			 add_args id new_args b,
			 na, b_option) cel,
	     List.map (fun (loc,cpl,e) -> (loc,cpl,add_args id new_args e)) cal
	    )
  | CLetTuple(loc,nal,(na,b_option),b1,b2) ->
      CLetTuple(loc,nal,(na,Option.map (add_args id new_args) b_option),
		add_args id new_args b1,
		add_args id new_args b2
	       )

  | CIf(loc,b1,(na,b_option),b2,b3) ->
      CIf(loc,add_args id new_args b1,
	  (na,Option.map (add_args id new_args) b_option),
	  add_args id new_args b2,
	  add_args id new_args b3
	 )
  | CHole _ -> b
  | CPatVar _ -> b
  | CEvar _ -> b
  | CSort _ -> b
  | CCast(loc,b1,b2)  ->
      CCast(loc,add_args id new_args b1,
	    Miscops.map_cast_type (add_args id new_args) b2)
  | CRecord (loc, pars) ->
      CRecord (loc, List.map (fun (e,o) -> e, add_args id new_args o) pars)
  | CNotation _ -> anomaly ~label:"add_args " (Pp.str "CNotation")
  | CGeneralization _ -> anomaly ~label:"add_args " (Pp.str "CGeneralization")
  | CPrim _ -> b
  | CDelimiters _ -> anomaly ~label:"add_args " (Pp.str "CDelimiters")
exception Stop of  Constrexpr.constr_expr


  [ chop_n_arrow n t ] chops the [ n ] first arrows in [ t ] 
    Acts on  

   Acts on Constrexpr.constr_expr
*)
let rec chop_n_arrow n t =
  if n <= 0
  then t (* If we have already removed all the arrows then return the type *)
  else (* If not we check the form of [t] *)
    match t with
      | Constrexpr.CProdN(_,nal_ta',t') -> (* If we have a forall, to result are possible :
					     either we need to discard more than the number of arrows contained
					     in this product declaration then we just recall [chop_n_arrow] on
					     the remaining number of arrow to chop and [t'] we discard it and
					     recall [chop_n_arrow], either this product contains more arrows
					     than the number we need to chop and then we return the new type
					  *)
	  begin
	    try
	      let new_n =
		let rec aux (n:int) = function
		    [] -> n
		| (nal,k,t'')::nal_ta' ->
		    let nal_l = List.length nal in
		    if n >= nal_l
		    then
		      aux (n - nal_l) nal_ta'
		    else
		      let new_t' =
			Constrexpr.CProdN(Loc.ghost,
					((snd (List.chop n nal)),k,t'')::nal_ta',t')
		      in
		      raise (Stop new_t')
		in
		aux n nal_ta'
	    in
	      chop_n_arrow new_n t'
	    with Stop t -> t
	  end
      | _ -> anomaly (Pp.str "Not enough products")


let rec get_args b t : Constrexpr.local_binder list *
    Constrexpr.constr_expr * Constrexpr.constr_expr =
  match b with
    | Constrexpr.CLambdaN (loc, (nal_ta), b') ->
	begin
	  let n =
	    (List.fold_left (fun n (nal,_,_) ->
			       n+List.length nal) 0 nal_ta )
	  in
	  let nal_tas,b'',t'' = get_args b' (chop_n_arrow n t) in
	  (List.map (fun (nal,k,ta) ->
		       (Constrexpr.LocalRawAssum (nal,k,ta))) nal_ta)@nal_tas, b'',t''
	end
    | _ -> [],b,t


let make_graph (f_ref:global_reference) =
  let c,c_body =
    match f_ref with
      | ConstRef c ->
	  begin try c,Global.lookup_constant c
	  with Not_found ->
	    raise (UserError ("",str "Cannot find " ++ Printer.pr_lconstr (mkConst c)) )
	  end
      | _ -> raise (UserError ("", str "Not a function reference") )
  in
  (match Global.body_of_constant_body c_body with
     | None -> error "Cannot build a graph over an axiom !"
     | Some body ->
	 let env = Global.env () in
	 let sigma = Evd.from_env env in
	 let extern_body,extern_type =
	   with_full_print (fun () ->
		(Constrextern.extern_constr false env sigma body,
		 Constrextern.extern_type false env sigma
FIXME
		)
	     )
	     ()
	 in
	 let (nal_tas,b,t)  = get_args extern_body extern_type in
	 let expr_list =
	   match b with
	     | Constrexpr.CFix(loc,l_id,fixexprl) ->
		 let l =
		   List.map
		     (fun (id,(n,recexp),bl,t,b) ->
			let loc, rec_id = Option.get n in
			let new_args =
			  List.flatten
			    (List.map
			       (function
				  | Constrexpr.LocalRawDef (na,_)-> []
				  | Constrexpr.LocalRawAssum (nal,_,_) ->
				      List.map
					(fun (loc,n) ->
					   CRef(Libnames.Ident(loc, Nameops.out_name n),None))
					nal
                                  | Constrexpr.LocalPattern _ -> assert false
			       )
			       nal_tas
			    )
			in
			let b' = add_args (snd id) new_args b in
			((((id,None), ( Some (Loc.ghost,rec_id),CStructRec),nal_tas@bl,t,Some b'),[]):(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list))
		     )
		     fixexprl
		 in
		 l
	     | _ ->
		let id = Label.to_id (con_label c) in
		 [(((Loc.ghost,id),None),(None,Constrexpr.CStructRec),nal_tas,t,Some b),[]]
	 in
	 let mp,dp,_ = repr_con c in
	 do_generate_principle [c,Univ.Instance.empty] error_error  false false expr_list;
	 (* We register the infos *)
	 List.iter
	   (fun ((((_,id),_),_,_,_,_),_) -> add_Function false (make_con mp dp (Label.of_id id)))
	   expr_list)

let do_generate_principle = do_generate_principle [] warning_error true



 then we get the principle 
i If there is not default lemma defined then,
			  we cross our finger and try to find a lemma named f_ind
			  (or f_rec, f_rect) i
 This one is neither defined ! 
 Declare local notations 
 Checks whether or not the mutual bloc is recursive 
 Assume that no `{ ... } contexts occur 
 We then register the Inductive graphs of the functions  
i The next call to mk_rel_id is valid since we have just construct the graph
	   Ensures by : do_built
	i
 funs_kn 
 No proof done 
 ok all the expressions are structural 
 If we have already removed all the arrows then return the type 
 If not we check the form of [t] 
 If we have a forall, to result are possible :
					     either we need to discard more than the number of arrows contained
					     in this product declaration then we just recall [chop_n_arrow] on
					     the remaining number of arrow to chop and [t'] we discard it and
					     recall [chop_n_arrow], either this product contains more arrows
					     than the number we need to chop and then we return the new type
					  
 We register the infos 
open Context.Rel.Declaration
open CErrors
open Util
open Names
open Term
open Pp
open Indfun_common
open Libnames
open Globnames
open Glob_term
open Declarations
open Misctypes
open Decl_kinds
open Sigma.Notations

let is_rec_info scheme_info =
  let test_branche min acc decl =
    acc || (
      let new_branche =
	it_mkProd_or_LetIn mkProp (fst (decompose_prod_assum (get_type decl))) in
      let free_rels_in_br = Termops.free_rels new_branche in
      let max = min + scheme_info.Tactics.npredicates in
      Int.Set.exists (fun i -> i >= min && i< max) free_rels_in_br
    )
  in
  List.fold_left_i test_branche 1 false (List.rev scheme_info.Tactics.branches)

let choose_dest_or_ind scheme_info =
  Tactics.induction_destruct (is_rec_info scheme_info) false

let functional_induction with_clean c princl pat =
  let res =
    let f,args = decompose_app c in
    fun g ->
    let princ,bindings, princ_type,g' =
      match princl with
  No principle is given let 's find the good one
	 begin
	   match kind_of_term f with
	   | Const (c',u) ->
	      let princ_option =
  we first try to find out a graph on f
		  try find_Function_infos c'
		  with Not_found ->
		    errorlabstrm "" (str "Cannot find induction information on "++
				       Printer.pr_lconstr (mkConst c') )
		in
		match Tacticals.elimination_sort_of_goal g with
		| InProp -> finfo.prop_lemma
		| InSet -> finfo.rec_lemma
		| InType -> finfo.rect_lemma
	      in
		try
		  let g',princ =
		    Tacmach.pf_eapply (Evd.fresh_global) g  (Globnames.ConstRef (Option.get princ_option )) in
		  princ,g'
		with Option.IsNone ->
		  let princ_name =
		    Indrec.make_elimination_ident
		      (Label.to_id (con_label c'))
		      (Tacticals.elimination_sort_of_goal g)
		  in
		  try
		    let princ_ref = const_of_id princ_name in
		    let (a,b) = Tacmach.pf_eapply (Evd.fresh_global) g princ_ref in
		    (b,a)
  mkConst(const_of_id princ_name ) ,  ( \ * FIXME * \ )
		    errorlabstrm "" (str "Cannot find induction principle for "
				     ++Printer.pr_lconstr (mkConst c') )
	      in
	      (princ,NoBindings, Tacmach.pf_unsafe_type_of g' princ,g')
	   | _ -> raise (UserError("",str "functional induction must be used with a function" ))
	 end
      | Some ((princ,binding)) ->
	 princ,binding,Tacmach.pf_unsafe_type_of g princ,g
    in
    let princ_infos = Tactics.compute_elim_sig princ_type in
    let args_as_induction_constr =
      let c_list =
	if princ_infos.Tactics.farg_in_concl
	then [c] else []
      in
      let encoded_pat_as_patlist =
        List.make (List.length args + List.length c_list - 1) None @ [pat] in
      List.map2 (fun c pat -> ((None,Tacexpr.ElimOnConstr ({ Tacexpr.delayed = fun env sigma -> Sigma ((c,NoBindings), sigma, Sigma.refl) })),(None,pat),None))
        (args@c_list) encoded_pat_as_patlist
    in
    let princ' = Some (princ,bindings) in
    let princ_vars =
      List.fold_right
	(fun a acc -> try Id.Set.add (destVar a) acc with DestKO -> acc)
	args
	Id.Set.empty
    in
    let old_idl = List.fold_right Id.Set.add (Tacmach.pf_ids_of_hyps g) Id.Set.empty in
    let old_idl = Id.Set.diff old_idl princ_vars in
    let subst_and_reduce g =
      if with_clean
      then
	let idl =
	  List.filter (fun id -> not (Id.Set.mem id old_idl))
	    (Tacmach.pf_ids_of_hyps g)
	in
	let flag =
	  Genredexpr.Cbv
	    {Redops.all_flags
	     with Genredexpr.rDelta = false;
	    }
	in
	Tacticals.tclTHEN
	  (Tacticals.tclMAP (fun id -> Tacticals.tclTRY (Proofview.V82.of_tactic (Equality.subst_gen (do_rewrite_dependent ()) [id]))) idl )
	  (Proofview.V82.of_tactic (Tactics.reduce flag Locusops.allHypsAndConcl))
	  g
      else Tacticals.tclIDTAC g
    in
    Tacticals.tclTHEN
      (Proofview.V82.of_tactic (choose_dest_or_ind
	 princ_infos
	 (args_as_induction_constr,princ')))
      subst_and_reduce
      g'
  in res

let rec abstract_glob_constr c = function
  | [] -> c
  | Constrexpr.LocalRawDef (x,b)::bl -> Constrexpr_ops.mkLetInC(x,b,abstract_glob_constr c bl)
  | Constrexpr.LocalRawAssum (idl,k,t)::bl ->
      List.fold_right (fun x b -> Constrexpr_ops.mkLambdaC([x],k,t,b)) idl
        (abstract_glob_constr c bl)
  | Constrexpr.LocalPattern _::bl -> assert false

let interp_casted_constr_with_implicits env sigma impls c  =
  Constrintern.intern_gen Pretyping.WithoutTypeConstraint env ~impls
    ~allow_patvar:false c


    Construct a fixpoint as a Glob_term 
    and not as a constr 

   Construct a fixpoint as a Glob_term
   and not as a constr
*)

let build_newrecursive
    lnameargsardef  =
  let env0 = Global.env() in
  let sigma = Evd.from_env env0 in
  let (rec_sign,rec_impls) =
    List.fold_left
      (fun (env,impls) (((_,recname),_),bl,arityc,_) ->
        let arityc = Constrexpr_ops.prod_constr_expr arityc bl in
        let arity,ctx = Constrintern.interp_type env0 sigma arityc in
	let evdref = ref (Evd.from_env env0) in
	let _, (_, impls') = Constrintern.interp_context_evars env evdref bl in
	let impl = Constrintern.compute_internalization_data env0 Constrintern.Recursive arity impls' in
        let open Context.Named.Declaration in
        (Environ.push_named (LocalAssum (recname,arity)) env, Id.Map.add recname impl impls))
      (env0,Constrintern.empty_internalization_env) lnameargsardef in
  let recdef =
    let f (_,bl,_,def) =
      let def = abstract_glob_constr def bl in
      interp_casted_constr_with_implicits
        rec_sign sigma rec_impls def
    in
    States.with_state_protection (List.map f) lnameargsardef
  in
  recdef,rec_impls

let build_newrecursive l = 
  let l' = List.map 
    (fun ((fixna,_,bll,ar,body_opt),lnot) -> 
       match body_opt with 
	 | Some body -> 
	     (fixna,bll,ar,body)
	 | None -> user_err_loc (Loc.ghost,"Function",str "Body of Function must be given")
    ) l
  in
  build_newrecursive l'

let is_rec names =
  let names = List.fold_right Id.Set.add names Id.Set.empty in
  let check_id id names =  Id.Set.mem id names in
  let rec lookup names = function
    | GVar(_,id) -> check_id id names
    | GRef _ | GEvar _ | GPatVar _ | GSort _ |  GHole _ -> false
    | GCast(_,b,_) -> lookup names b
    | GRec _ -> error "GRec not handled"
    | GIf(_,b,_,lhs,rhs) ->
	(lookup names b) || (lookup names lhs) || (lookup names rhs)
    | GLetIn(_,na,t,b) | GLambda(_,na,_,t,b) | GProd(_,na,_,t,b)  ->
	lookup names t || lookup (Nameops.name_fold Id.Set.remove na names) b
    | GLetTuple(_,nal,_,t,b) -> lookup names t ||
	lookup
	  (List.fold_left
	     (fun acc na -> Nameops.name_fold Id.Set.remove na acc)
	     names
	     nal
	  )
	  b
    | GApp(_,f,args) -> List.exists (lookup names) (f::args)
    | GCases(_,_,_,el,brl) ->
	List.exists (fun (e,_) -> lookup names e) el ||
	  List.exists (lookup_br names) brl
  and lookup_br names (_,idl,_,rt) =
    let new_names = List.fold_right Id.Set.remove idl names in
    lookup new_names rt
  in
  lookup names

let rec local_binders_length = function
  | [] -> 0
  | Constrexpr.LocalRawDef _::bl -> 1 + local_binders_length bl
  | Constrexpr.LocalRawAssum (idl,_,_)::bl -> List.length idl + local_binders_length bl
  | Constrexpr.LocalPattern _::bl -> assert false

let prepare_body ((name,_,args,types,_),_) rt =
  let n = local_binders_length args in
    Pp.msgnl ( str " nb lambda to chop : " + + str ( string_of_int n ) + + fnl ( ) + + Printer.pr_glob_constr rt ) ;
  let fun_args,rt' = chop_rlambda_n n rt in
  (fun_args,rt')

let process_vernac_interp_error e =
  fst (ExplainErr.process_vernac_interp_error (e, Exninfo.null))

let warn_funind_cannot_build_inversion =
  CWarnings.create ~name:"funind-cannot-build-inversion" ~category:"funind"
    (fun e' -> strbrk "Cannot build inversion information" ++
                 if do_observe () then (fnl() ++ CErrors.print e') else mt ())

let derive_inversion fix_names =
  try
    let evd' = Evd.from_env (Global.env ()) in 
  we first transform the fix_names identifier into their corresponding constant
    let evd',fix_names_as_constant =
      List.fold_right
	(fun id (evd,l) ->
	 let evd,c =
	   Evd.fresh_global
	     (Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident id)) in 
	 evd, destConst c::l
	)
	fix_names
	(evd',[])
    in

        Then we check that the graphs have been defined 
        If one of the graphs have n't been defined 
        we do nothing 
    
       Then we check that the graphs have been defined
       If one of the graphs haven't been defined
       we do nothing
    *)
    List.iter (fun c -> ignore (find_Function_infos (fst c))) fix_names_as_constant ;
    try
      let evd', lind =
	List.fold_right
	  (fun id (evd,l) ->
	   let evd,id = 
	     Evd.fresh_global
	       (Global.env ()) evd
	       (Constrintern.locate_reference (Libnames.qualid_of_ident (mk_rel_id id)))
	   in 
	   evd,(fst (destInd id))::l
	  )
	  fix_names
	  (evd',[])
      in
      Invfun.derive_correctness
	Functional_principles_types.make_scheme
	functional_induction
	fix_names_as_constant
	lind;
      with e when CErrors.noncritical e ->
      let e' = process_vernac_interp_error e in
      warn_funind_cannot_build_inversion e'
  with e when CErrors.noncritical e ->
    let e' = process_vernac_interp_error e in
    warn_funind_cannot_build_inversion e'

let warn_cannot_define_graph =
  CWarnings.create ~name:"funind-cannot-define-graph" ~category:"funind"
    (fun (names,error) -> strbrk "Cannot define graph(s) for " ++
      h 1 names ++ error)

let warn_cannot_define_principle =
  CWarnings.create ~name:"funind-cannot-define-principle" ~category:"funind"
  (fun (names,error) -> strbrk "Cannot define induction principle(s) for "++
      h 1 names ++ error)

let warning_error names e =
  let e = process_vernac_interp_error e in
  let e_explain e =
    match e with
      | ToShow e -> 
	let e = process_vernac_interp_error e in
	spc () ++ CErrors.print e
      | _ -> 
	if do_observe () 
	then 
	  let e = process_vernac_interp_error e in 
	  (spc () ++ CErrors.print e)
	else mt ()
  in
  match e with
    | Building_graph e ->
       let names = prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names in
       warn_cannot_define_graph (names,e_explain e)
    | Defining_principle e ->
       let names = prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names in
       warn_cannot_define_principle (names,e_explain e)
    | _ -> raise e

let error_error names e =
  let e = process_vernac_interp_error e in
  let e_explain e =
    match e with
      | ToShow e -> spc () ++ CErrors.print e
      | _ -> if do_observe () then (spc () ++ CErrors.print e) else mt ()
  in
  match e with
    | Building_graph e ->
	errorlabstrm ""
	  (str "Cannot define graph(s) for " ++
	     h 1 (prlist_with_sep (fun _ -> str","++spc ()) Ppconstr.pr_id names) ++
	     e_explain e)
    | _ -> raise e

let generate_principle (evd:Evd.evar_map ref) pconstants on_error
    is_general do_built (fix_rec_l:(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list) list) recdefs  interactive_proof
    (continue_proof : int -> Names.constant array -> Term.constr array -> int ->
      Tacmach.tactic) : unit =
  let names = List.map (function (((_, name),_),_,_,_,_),_ -> name) fix_rec_l in
  let fun_bodies = List.map2 prepare_body fix_rec_l recdefs in
  let funs_args = List.map fst fun_bodies in
  let funs_types =  List.map (function ((_,_,_,types,_),_) -> types) fix_rec_l in
  try
    Glob_term_to_relation.build_inductive !evd pconstants funs_args funs_types recdefs;
    if do_built
    then
      begin
	let f_R_mut = Ident (Loc.ghost,mk_rel_id (List.nth names 0)) in
	let ind_kn =
	  fst (locate_with_msg
		 (pr_reference f_R_mut++str ": Not an inductive type!")
		 locate_ind
		 f_R_mut)
	in
	let fname_kn (((fname,_),_,_,_,_),_) =
	  let f_ref = Ident fname in
	  locate_with_msg
	    (pr_reference f_ref++str ": Not an inductive type!")
	    locate_constant
	    f_ref
	in
	let funs_kn = Array.of_list (List.map fname_kn fix_rec_l) in
	let _ =
	  List.map_i
	    (fun i x ->
	     let princ = Indrec.lookup_eliminator (ind_kn,i) (InProp) in
	     let env = Global.env () in
	     let evd = ref (Evd.from_env env) in
	     let evd',uprinc = Evd.fresh_global env !evd princ in
             let _ = evd := evd' in 
	     let princ_type = Typing.e_type_of ~refresh:true env evd uprinc in
    	     Functional_principles_types.generate_functional_principle
	       evd
		 interactive_proof
		 princ_type
		 None
		 None
		 (Array.of_list pconstants)
		 i
		 (continue_proof  0 [|funs_kn.(i)|])
	    )
	    0
	    fix_rec_l
	in
	Array.iter (add_Function is_general) funs_kn;
	()
      end
  with e when CErrors.noncritical e ->
    on_error names e

let register_struct is_rec (fixpoint_exprl:(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list) list) =
  match fixpoint_exprl with
    | [(((_,fname),pl),_,bl,ret_type,body),_] when not is_rec ->
      let body = match body with | Some body -> body | None -> user_err_loc (Loc.ghost,"Function",str "Body of Function must be given") in 
      Command.do_definition
	fname
	(Decl_kinds.Global,(Flags.is_universe_polymorphism ()),Decl_kinds.Definition) pl
	bl None body (Some ret_type) (Lemmas.mk_hook (fun _ _ -> ()));
       let evd,rev_pconstants =
	 List.fold_left
	   (fun (evd,l) ((((_,fname),_),_,_,_,_),_) ->
	    let evd,c =
	      Evd.fresh_global
		(Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident fname)) in
	    evd,((destConst c)::l)
	   )
	   (Evd.from_env (Global.env ()),[])
	   fixpoint_exprl
       in
       evd,List.rev rev_pconstants
    | _ ->
       Command.do_fixpoint Global (Flags.is_universe_polymorphism ()) fixpoint_exprl;
       let evd,rev_pconstants =
	 List.fold_left
	   (fun (evd,l) ((((_,fname),_),_,_,_,_),_) ->
	    let evd,c =
	      Evd.fresh_global
		(Global.env ()) evd (Constrintern.locate_reference (Libnames.qualid_of_ident fname)) in
	    evd,((destConst c)::l)
	   )
	   (Evd.from_env (Global.env ()),[])
	   fixpoint_exprl
       in
       evd,List.rev rev_pconstants
	   

let generate_correction_proof_wf f_ref tcc_lemma_ref
    is_mes functional_ref eq_ref rec_arg_num rec_arg_type nb_args relation
    (_: int) (_:Names.constant array) (_:Term.constr array) (_:int) : Tacmach.tactic =
  Functional_principles_proofs.prove_principle_for_gen
    (f_ref,functional_ref,eq_ref)
    tcc_lemma_ref is_mes  rec_arg_num rec_arg_type relation


let register_wf ?(is_mes=false) fname rec_impls wf_rel_expr wf_arg using_lemmas args ret_type body
    pre_hook
    =
  let type_of_f = Constrexpr_ops.prod_constr_expr ret_type args in
  let rec_arg_num =
    let names =
      List.map
	snd
	(Constrexpr_ops.names_of_local_assums args)
    in
    match wf_arg with
      | None ->
	if Int.equal (List.length names) 1 then 1
	else error "Recursive argument must be specified"
      | Some wf_arg ->
	  List.index Name.equal (Name wf_arg) names
  in
  let unbounded_eq =
    let f_app_args =
      Constrexpr.CAppExpl
	(Loc.ghost,
	 (None,(Ident (Loc.ghost,fname)),None) ,
	 (List.map
	    (function
	       | _,Anonymous -> assert false
	       | _,Name e -> (Constrexpr_ops.mkIdentC e)
	    )
	    (Constrexpr_ops.names_of_local_assums args)
	 )
	)
    in
    Constrexpr.CApp (Loc.ghost,(None,Constrexpr_ops.mkRefC (Qualid (Loc.ghost,(qualid_of_string "Logic.eq")))),
		    [(f_app_args,None);(body,None)])
  in
  let eq = Constrexpr_ops.prod_constr_expr unbounded_eq args in
  let hook ((f_ref,_) as fconst) tcc_lemma_ref (functional_ref,_) (eq_ref,_) rec_arg_num rec_arg_type
      nb_args relation =
    try
      pre_hook [fconst]
	(generate_correction_proof_wf f_ref tcc_lemma_ref is_mes
	   functional_ref eq_ref rec_arg_num rec_arg_type nb_args relation
	);
      derive_inversion [fname]
    with e when CErrors.noncritical e ->
      ()
  in
  Recdef.recursive_definition
    is_mes fname rec_impls
    type_of_f
    wf_rel_expr
    rec_arg_num
    eq
    hook
    using_lemmas


let register_mes fname rec_impls wf_mes_expr wf_rel_expr_opt wf_arg using_lemmas args ret_type body =
  let wf_arg_type,wf_arg =
    match wf_arg with
      | None ->
	  begin
	    match args with
	      | [Constrexpr.LocalRawAssum ([(_,Name x)],k,t)] -> t,x
	      | _ -> error "Recursive argument must be specified"
	  end
      | Some wf_args ->
	  try
	    match
	      List.find
		(function
		   | Constrexpr.LocalRawAssum(l,k,t) ->
		       List.exists
			 (function (_,Name id) -> Id.equal id wf_args | _ -> false)
			 l
		   | _ -> false
		)
		args
	    with
	      | Constrexpr.LocalRawAssum(_,k,t)  ->	    t,wf_args
	      | _ -> assert false
	  with Not_found -> assert false
  in
  let wf_rel_from_mes,is_mes = 
    match wf_rel_expr_opt with 
      | None ->
	  let ltof =
	    let make_dir l = DirPath.make (List.rev_map Id.of_string l) in
	    Libnames.Qualid (Loc.ghost,Libnames.qualid_of_path
			       (Libnames.make_path (make_dir ["Arith";"Wf_nat"]) (Id.of_string "ltof")))
	  in
	  let fun_from_mes =
	    let applied_mes =
	      Constrexpr_ops.mkAppC(wf_mes_expr,[Constrexpr_ops.mkIdentC wf_arg])    in
	    Constrexpr_ops.mkLambdaC ([(Loc.ghost,Name wf_arg)],Constrexpr_ops.default_binder_kind,wf_arg_type,applied_mes)
	  in
	  let wf_rel_from_mes =
	    Constrexpr_ops.mkAppC(Constrexpr_ops.mkRefC  ltof,[wf_arg_type;fun_from_mes])
	  in
	  wf_rel_from_mes,true
      | Some wf_rel_expr -> 
	  let wf_rel_with_mes = 
	    let a = Names.Id.of_string "___a" in 
	    let b = Names.Id.of_string "___b" in 
	    Constrexpr_ops.mkLambdaC(
	      [Loc.ghost,Name a;Loc.ghost,Name b],
	      Constrexpr.Default Explicit,
	      wf_arg_type,
	      Constrexpr_ops.mkAppC(wf_rel_expr,
			       [
				Constrexpr_ops.mkAppC(wf_mes_expr,[Constrexpr_ops.mkIdentC a]);
				Constrexpr_ops.mkAppC(wf_mes_expr,[Constrexpr_ops.mkIdentC b])
			       ])
			      )
	  in
	  wf_rel_with_mes,false
  in			       
  register_wf ~is_mes:is_mes fname rec_impls wf_rel_from_mes (Some wf_arg)
    using_lemmas args ret_type body

let map_option f = function 
  | None -> None 
  | Some v -> Some (f v)

open Constrexpr
open Topconstr

let make_assoc assoc l1 l2 =
  let fold assoc a b = match a, b with
  | (_, Name na), (_, Name id) -> Id.Map.add na id assoc
  | _, _ -> assoc
  in
  List.fold_left2 fold assoc l1 l2

let rec rebuild_bl (aux,assoc) bl typ = 
	match bl,typ with 
	  | [], _ -> (List.rev aux,replace_vars_constr_expr assoc typ,assoc)
	  | (Constrexpr.LocalRawAssum(nal,bk,_))::bl',typ ->
	     rebuild_nal (aux,assoc) bk bl' nal (List.length nal) typ
	  | (Constrexpr.LocalRawDef(na,_))::bl',Constrexpr.CLetIn(_,_,nat,typ') ->
	    rebuild_bl ((Constrexpr.LocalRawDef(na,replace_vars_constr_expr assoc nat)::aux),assoc)
	      bl' typ'
	  | _ -> assert false
      and rebuild_nal (aux,assoc) bk bl' nal lnal typ = 
	match nal,typ with 
	  | [], _ -> rebuild_bl (aux,assoc) bl' typ
	  | _,CProdN(_,[],typ) -> rebuild_nal (aux,assoc) bk bl' nal lnal typ
	  | _,CProdN(_,(nal',bk',nal't)::rest,typ') -> 
	    let lnal' = List.length nal' in 
	    if lnal' >= lnal 
	    then 
	      let old_nal',new_nal' = List.chop lnal nal' in
	      let nassoc = make_assoc assoc old_nal' nal in
	      let assum = LocalRawAssum(nal,bk,replace_vars_constr_expr assoc nal't) in
	      rebuild_bl ((assum :: aux), nassoc) bl' 
		(if List.is_empty new_nal' && List.is_empty rest
		 then typ'
		 else if List.is_empty new_nal'
		 then CProdN(Loc.ghost,rest,typ')
		 else CProdN(Loc.ghost,((new_nal',bk',nal't)::rest),typ'))
	    else 
	      let captured_nal,non_captured_nal = List.chop lnal' nal in
	      let nassoc = make_assoc assoc nal' captured_nal in
	      let assum = LocalRawAssum(captured_nal,bk,replace_vars_constr_expr assoc nal't) in
	      rebuild_nal ((assum :: aux), nassoc)
		bk bl' non_captured_nal (lnal - lnal') (CProdN(Loc.ghost,rest,typ'))
	  | _ -> assert false

let rebuild_bl (aux,assoc) bl typ = rebuild_bl (aux,assoc) bl typ

let recompute_binder_list (fixpoint_exprl : (Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list) list) = 
  let fixl,ntns = Command.extract_fixpoint_components false fixpoint_exprl in
  let ((_,_,typel),_,ctx,_) = Command.interp_fixpoint fixl ntns in
  let constr_expr_typel = 
    with_full_print (List.map (Constrextern.extern_constr false (Global.env ()) (Evd.from_ctx ctx))) typel in
  let fixpoint_exprl_with_new_bl = 
    List.map2 (fun ((lna,(rec_arg_opt,rec_order),bl,ret_typ,opt_body),notation_list) fix_typ -> 
     
      let new_bl',new_ret_type,_ = rebuild_bl ([],Id.Map.empty) bl fix_typ in 
      (((lna,(rec_arg_opt,rec_order),new_bl',new_ret_type,opt_body),notation_list):(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list))
    )
      fixpoint_exprl constr_expr_typel 
  in 
  fixpoint_exprl_with_new_bl
  

let do_generate_principle pconstants on_error register_built interactive_proof
    (fixpoint_exprl:(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list) list) :unit =
  List.iter (fun (_,l) -> if not (List.is_empty l) then error "Function does not support notations for now") fixpoint_exprl;
  let _is_struct =
    match fixpoint_exprl with
      | [((_,(wf_x,Constrexpr.CWfRec wf_rel),_,_,_),_) as fixpoint_expr] ->
 	  let (((((_,name),pl),_,args,types,body)),_)  as fixpoint_expr =
	    match recompute_binder_list [fixpoint_expr] with 
	      | [e] -> e
	      | _ -> assert false
	  in 
	  let fixpoint_exprl = [fixpoint_expr] in 
	  let body = match body with | Some body -> body | None -> user_err_loc (Loc.ghost,"Function",str "Body of Function must be given") in 
	  let recdefs,rec_impls = build_newrecursive fixpoint_exprl in
	  let using_lemmas = [] in 
	  let pre_hook pconstants =
	    generate_principle
	      (ref (Evd.from_env (Global.env ())))
	      pconstants
	      on_error
	      true
	      register_built
	      fixpoint_exprl
	      recdefs
	      true
	  in
	  if register_built
	  then register_wf name rec_impls wf_rel (map_option snd wf_x) using_lemmas args types body pre_hook;
	  false
      |[((_,(wf_x,Constrexpr.CMeasureRec(wf_mes,wf_rel_opt)),_,_,_),_) as fixpoint_expr] ->
 	  let (((((_,name),_),_,args,types,body)),_)  as fixpoint_expr =
	    match recompute_binder_list [fixpoint_expr] with 
	      | [e] -> e
	      | _ -> assert false
	  in 
	  let fixpoint_exprl = [fixpoint_expr] in 
	  let recdefs,rec_impls = build_newrecursive fixpoint_exprl in
	  let using_lemmas = [] in
	  let body = match body with | Some body -> body | None -> user_err_loc (Loc.ghost,"Function",str "Body of Function must be given") in 
	  let pre_hook pconstants =
	    generate_principle
	      (ref (Evd.from_env (Global.env ())))
	      pconstants
	      on_error
	      true
	      register_built
	      fixpoint_exprl
	      recdefs
	      true
	  in
	  if register_built
	  then register_mes name rec_impls wf_mes wf_rel_opt (map_option snd wf_x) using_lemmas args types body pre_hook;
	  true
      | _ ->
	  List.iter (function ((_na,(_,ord),_args,_body,_type),_not) ->
		       match ord with 
			 | Constrexpr.CMeasureRec _ | Constrexpr.CWfRec _ ->
			     error
			       ("Cannot use mutual definition with well-founded recursion or measure")
			 | _ -> ()
		    )
	    fixpoint_exprl;
	let fixpoint_exprl = recompute_binder_list fixpoint_exprl in 
	let fix_names =
	  List.map (function ((((_,name),_),_,_,_,_),_) -> name) fixpoint_exprl
	in
  	let recdefs,rec_impls = build_newrecursive fixpoint_exprl in
	let is_rec = List.exists (is_rec fix_names) recdefs in
	let evd,pconstants =
	  if register_built
	  then register_struct is_rec fixpoint_exprl
	  else (Evd.from_env (Global.env ()),pconstants)
	in
	let evd = ref evd in 
	generate_principle
	  (ref !evd)
	  pconstants
	  on_error
	  false
	  register_built
	  fixpoint_exprl
	  recdefs
	  interactive_proof
	  (Functional_principles_proofs.prove_princ_for_struct evd interactive_proof);
	if register_built then begin derive_inversion fix_names; end;
	true; 
  in
  ()

let rec add_args id new_args b =
  match b with
  | CRef (r,_) ->
      begin      match r with
	| Libnames.Ident(loc,fname) when Id.equal fname id ->
	    CAppExpl(Loc.ghost,(None,r,None),new_args)
	| _ -> b
      end
  | CFix  _  | CCoFix _ -> anomaly ~label:"add_args " (Pp.str "todo")
  | CProdN(loc,nal,b1) ->
      CProdN(loc,
	     List.map (fun (nal,k,b2) -> (nal,k,add_args id new_args b2)) nal,
	     add_args id new_args  b1)
  | CLambdaN(loc,nal,b1) ->
      CLambdaN(loc,
	       List.map (fun (nal,k,b2) -> (nal,k,add_args id new_args  b2)) nal,
	       add_args id new_args  b1)
  | CLetIn(loc,na,b1,b2) ->
      CLetIn(loc,na,add_args id new_args b1,add_args id new_args b2)
  | CAppExpl(loc,(pf,r,us),exprl) ->
      begin
	match r with
	| Libnames.Ident(loc,fname) when Id.equal fname id ->
	    CAppExpl(loc,(pf,r,us),new_args@(List.map (add_args id new_args) exprl))
	| _ -> CAppExpl(loc,(pf,r,us),List.map (add_args id new_args) exprl)
      end
  | CApp(loc,(pf,b),bl) ->
      CApp(loc,(pf,add_args id new_args b),
	   List.map (fun (e,o) -> add_args id new_args e,o) bl)
  | CCases(loc,sty,b_option,cel,cal) ->
      CCases(loc,sty,Option.map (add_args id new_args) b_option,
	     List.map (fun (b,na,b_option) ->
			 add_args id new_args b,
			 na, b_option) cel,
	     List.map (fun (loc,cpl,e) -> (loc,cpl,add_args id new_args e)) cal
	    )
  | CLetTuple(loc,nal,(na,b_option),b1,b2) ->
      CLetTuple(loc,nal,(na,Option.map (add_args id new_args) b_option),
		add_args id new_args b1,
		add_args id new_args b2
	       )

  | CIf(loc,b1,(na,b_option),b2,b3) ->
      CIf(loc,add_args id new_args b1,
	  (na,Option.map (add_args id new_args) b_option),
	  add_args id new_args b2,
	  add_args id new_args b3
	 )
  | CHole _ -> b
  | CPatVar _ -> b
  | CEvar _ -> b
  | CSort _ -> b
  | CCast(loc,b1,b2)  ->
      CCast(loc,add_args id new_args b1,
	    Miscops.map_cast_type (add_args id new_args) b2)
  | CRecord (loc, pars) ->
      CRecord (loc, List.map (fun (e,o) -> e, add_args id new_args o) pars)
  | CNotation _ -> anomaly ~label:"add_args " (Pp.str "CNotation")
  | CGeneralization _ -> anomaly ~label:"add_args " (Pp.str "CGeneralization")
  | CPrim _ -> b
  | CDelimiters _ -> anomaly ~label:"add_args " (Pp.str "CDelimiters")
exception Stop of  Constrexpr.constr_expr


  [ chop_n_arrow n t ] chops the [ n ] first arrows in [ t ] 
    Acts on  

   Acts on Constrexpr.constr_expr
*)
let rec chop_n_arrow n t =
  if n <= 0
    match t with
	  begin
	    try
	      let new_n =
		let rec aux (n:int) = function
		    [] -> n
		| (nal,k,t'')::nal_ta' ->
		    let nal_l = List.length nal in
		    if n >= nal_l
		    then
		      aux (n - nal_l) nal_ta'
		    else
		      let new_t' =
			Constrexpr.CProdN(Loc.ghost,
					((snd (List.chop n nal)),k,t'')::nal_ta',t')
		      in
		      raise (Stop new_t')
		in
		aux n nal_ta'
	    in
	      chop_n_arrow new_n t'
	    with Stop t -> t
	  end
      | _ -> anomaly (Pp.str "Not enough products")


let rec get_args b t : Constrexpr.local_binder list *
    Constrexpr.constr_expr * Constrexpr.constr_expr =
  match b with
    | Constrexpr.CLambdaN (loc, (nal_ta), b') ->
	begin
	  let n =
	    (List.fold_left (fun n (nal,_,_) ->
			       n+List.length nal) 0 nal_ta )
	  in
	  let nal_tas,b'',t'' = get_args b' (chop_n_arrow n t) in
	  (List.map (fun (nal,k,ta) ->
		       (Constrexpr.LocalRawAssum (nal,k,ta))) nal_ta)@nal_tas, b'',t''
	end
    | _ -> [],b,t


let make_graph (f_ref:global_reference) =
  let c,c_body =
    match f_ref with
      | ConstRef c ->
	  begin try c,Global.lookup_constant c
	  with Not_found ->
	    raise (UserError ("",str "Cannot find " ++ Printer.pr_lconstr (mkConst c)) )
	  end
      | _ -> raise (UserError ("", str "Not a function reference") )
  in
  (match Global.body_of_constant_body c_body with
     | None -> error "Cannot build a graph over an axiom !"
     | Some body ->
	 let env = Global.env () in
	 let sigma = Evd.from_env env in
	 let extern_body,extern_type =
	   with_full_print (fun () ->
		(Constrextern.extern_constr false env sigma body,
		 Constrextern.extern_type false env sigma
FIXME
		)
	     )
	     ()
	 in
	 let (nal_tas,b,t)  = get_args extern_body extern_type in
	 let expr_list =
	   match b with
	     | Constrexpr.CFix(loc,l_id,fixexprl) ->
		 let l =
		   List.map
		     (fun (id,(n,recexp),bl,t,b) ->
			let loc, rec_id = Option.get n in
			let new_args =
			  List.flatten
			    (List.map
			       (function
				  | Constrexpr.LocalRawDef (na,_)-> []
				  | Constrexpr.LocalRawAssum (nal,_,_) ->
				      List.map
					(fun (loc,n) ->
					   CRef(Libnames.Ident(loc, Nameops.out_name n),None))
					nal
                                  | Constrexpr.LocalPattern _ -> assert false
			       )
			       nal_tas
			    )
			in
			let b' = add_args (snd id) new_args b in
			((((id,None), ( Some (Loc.ghost,rec_id),CStructRec),nal_tas@bl,t,Some b'),[]):(Vernacexpr.fixpoint_expr * Vernacexpr.decl_notation list))
		     )
		     fixexprl
		 in
		 l
	     | _ ->
		let id = Label.to_id (con_label c) in
		 [(((Loc.ghost,id),None),(None,Constrexpr.CStructRec),nal_tas,t,Some b),[]]
	 in
	 let mp,dp,_ = repr_con c in
	 do_generate_principle [c,Univ.Instance.empty] error_error  false false expr_list;
	 List.iter
	   (fun ((((_,id),_),_,_,_,_),_) -> add_Function false (make_con mp dp (Label.of_id id)))
	   expr_list)

let do_generate_principle = do_generate_principle [] warning_error true


