(************************************************************************)
(*  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 Tacmach
open Proof_type
open Libobject
open Reductionops
open Term
open Termops
open Names
open Entries
open Libnames
open Nameops
open Util
open Pp
open Printer
open Environ
open Tactics 
open Tacticals
open Vernacexpr
open Safe_typing
open Nametab
open Decl_kinds
open Constrintern

let replace = ref (fun _ _ -> assert false)
let register_replace f = replace := f

type reflexive_relation =
   { refl_a: constr ;
     refl_aeq: constr;
     refl_refl: constr;
     refl_sym: constr option
   }

type 'a relation_class =
   ACReflexive of 'a    (* the relation of the reflexive_relation
                           or the reflexive_relation*)
 | ACLeibniz of constr  (* the carrier *)

type 'a morphism =
    { args : 'a relation_class list;
      output : 'a relation_class;
      lem : constr;
    }

type funct =
    { f_args : constr list;
      f_output : constr
    }

type morphism_class =
   ACMorphism of reflexive_relation morphism
 | ACFunction of funct

let subst_mps_in_relation_class subst =
 function
    ACReflexive  t -> ACReflexive  (subst_mps subst t)
  | ACLeibniz t -> ACLeibniz (subst_mps subst t) 

let constr_relation_class_of_relation_relation_class =
 function
    ACReflexive relation -> ACReflexive relation.refl_aeq
  | ACLeibniz t -> ACLeibniz t
 

let constr_of c = Constrintern.interp_constr Evd.empty (Global.env()) c

let constant dir s = Coqlib.gen_constant "Setoid_replace" ("Setoids"::dir) s
let init_constant dir s = Coqlib.gen_constant "Setoid_replace" ("Init"::dir) s
let reference dir s = Coqlib.gen_reference "Setoid_replace" ("Setoids"::dir) s
let eval_reference dir s = EvalConstRef (destConst (constant dir s))
let eval_init_reference dir s = EvalConstRef (destConst (init_constant dir s))

let current_constant id =
  try
    global_reference id
  with Not_found ->
    anomaly ("Setoid: cannot find " ^ (string_of_id id))

(* From Setoid.v *)

let coq_is_reflexive = lazy(constant ["Setoid"] "is_reflexive")

let coq_Relation_Class = lazy(constant ["Setoid"] "Relation_Class")
let coq_Setoid_Theory = lazy(constant ["Setoid"] "Setoid_Theory")
let coq_Morphism_Theory = lazy(constant ["Setoid"] "Morphism_Theory")
let coq_Build_Morphism_Theory= lazy(constant ["Setoid"] "Build_Morphism_Theory")

let coq_Reflexive = lazy(constant ["Setoid"] "Reflexive")
let coq_Leibniz = lazy(constant ["Setoid"] "Leibniz")

let coq_seq_refl = lazy(constant ["Setoid"] "Seq_refl")
let coq_seq_sym = lazy(constant ["Setoid"] "Seq_sym")
let coq_seq_trans = lazy(constant ["Setoid"] "Seq_trans")

let coq_singl = lazy(constant ["Setoid"] "singl")
let coq_cons = lazy(constant ["Setoid"] "cons")

let coq_equality_morphism_of_setoid_theory =
 lazy(constant ["Setoid"] "equality_morphism_of_setoid_theory")
let coq_make_compatibility_goal =
 lazy(constant ["Setoid"] "make_compatibility_goal")
let coq_make_compatibility_goal_ref =
 lazy(reference ["Setoid"] "make_compatibility_goal")
let coq_make_compatibility_goal_aux_ref =
 lazy(reference ["Setoid"] "make_compatibility_goal_aux")

let coq_App = lazy(constant ["Setoid"] "App")
let coq_Toreplace = lazy(constant ["Setoid"] "Toreplace")
let coq_Tokeep = lazy(constant ["Setoid"] "Tokeep")
let coq_Imp = lazy(constant ["Setoid"] "Imp")
let coq_fcl_singl = lazy(constant ["Setoid"] "fcl_singl")
let coq_fcl_cons = lazy(constant ["Setoid"] "fcl_cons")

let coq_setoid_rewrite = lazy(constant ["Setoid"] "setoid_rewrite")
let coq_proj2 = lazy(init_constant ["Logic"] "proj2")

let coq_morphism_theory_of_function =
 lazy(constant ["Setoid"] "morphism_theory_of_function")
let coq_morphism_theory_of_predicate =
 lazy(constant ["Setoid"] "morphism_theory_of_predicate")
let coq_relation_of_relation_class =
 lazy(eval_reference ["Setoid"] "relation_of_relation_class")
let coq_interp = lazy(eval_reference ["Setoid"] "interp")
let coq_Morphism_Context_rect2 =
 lazy(eval_reference ["Setoid"] "Morphism_Context_rect2")
let coq_iff = lazy(eval_init_reference ["Logic"] "iff")

let coq_prop_relation =
 lazy (
  ACReflexive {
     refl_a = mkProp ;
     refl_aeq = init_constant ["Logic"] "iff" ;
     refl_refl = init_constant ["Logic"] "iff_refl";
     refl_sym = Some (init_constant ["Logic"] "iff_sym")
   })


(************************* Table of declared relations **********************)


(* Relations are stored in a table which is synchronised with the Reset mechanism. *)

let relation_table = ref Gmap.empty

let relation_table_add (s,th) = relation_table := Gmap.add s th !relation_table
let relation_table_find s = Gmap.find s !relation_table
let relation_table_mem s = Gmap.mem s !relation_table

let prrelation s =
 str "(" ++ prterm s.refl_a ++ str "," ++ prterm s.refl_aeq ++ str ")"

let prrelation_class =
 function
    ACReflexive eq  ->
     (try prrelation (relation_table_find eq)
      with Not_found ->
       (*CSC: still "setoid" in the error message *)
       str "[[ Error: setoid on equality " ++ prterm eq ++ str " not found! ]]")
  | ACLeibniz ty -> prterm ty

let prmorphism k m =
  prterm k ++ str ": " ++
  prlist_with_sep (fun () -> str " -> ") prrelation_class m.args ++
  str " -> " ++ prrelation_class m.output


(* A function that gives back the only relation_class on a given carrier *)
(*CSC: this implementation is really inefficient. I should define a new
  map to make it efficient. However, is this really worth of? *)
let default_relation_for_carrier a =
 let rng = Gmap.rng !relation_table in
 match List.filter (fun {refl_a=refl_a} -> refl_a = a) rng with
    [] -> ACLeibniz a
  | relation::tl ->
     if tl <> [] then
      msg_warning
       (*CSC: still "setoid" in the error message *)
       (str "There are several setoids whose carrier is " ++ prterm a ++
         str ". The setoid " ++ prrelation relation ++
         str " is randomly chosen.") ;
     ACReflexive relation

let relation_morphism_of_constr_morphism =
 let relation_relation_class_of_constr_relation_class =
  function
     ACLeibniz t -> ACLeibniz t
   | ACReflexive aeq ->
      ACReflexive (try relation_table_find aeq with Not_found -> assert false)
 in
  function mor ->
   let args' = List.map relation_relation_class_of_constr_relation_class mor.args in
   let output' = relation_relation_class_of_constr_relation_class mor.output in
    {mor with args=args' ; output=output'}

let subst_relation subst relation = 
  let refl_a' = subst_mps subst relation.refl_a in
  let refl_aeq' = subst_mps subst relation.refl_aeq in
  let refl_refl' = subst_mps subst relation.refl_refl in
  let refl_sym' = option_app (subst_mps subst) relation.refl_sym in
    if refl_a' == relation.refl_a
      && refl_aeq' == relation.refl_aeq
      && refl_refl' == relation.refl_refl
      && refl_sym' == relation.refl_sym
    then
      relation
    else
      { refl_a = refl_a' ;
        refl_aeq = refl_aeq' ;
        refl_refl = refl_refl' ;
        refl_sym = refl_sym'
      }
      
let equiv_list () = List.map (fun x -> x.refl_aeq) (Gmap.rng !relation_table)

let _ = 
  Summary.declare_summary "relation-table"
    { Summary.freeze_function = (fun () -> !relation_table);
      Summary.unfreeze_function = (fun t -> relation_table := t);
      Summary.init_function = (fun () -> relation_table := Gmap .empty);
      Summary.survive_module = true;
      Summary.survive_section = false }

(* Declare a new type of object in the environment : "relation-theory". *)

let (relation_to_obj, obj_to_relation)=
  let cache_set (_,(s, th)) =
   let th' =
    if relation_table_mem s then
     begin
      let old_relation = relation_table_find s in
       let th' =
        {th with refl_sym =
          match th.refl_sym with
             None -> old_relation.refl_sym
           | Some t -> Some t} in
        (*CSC: still "setoid" in the error message *)
        msg_warning
         (str "The setoid " ++ prrelation th' ++ str " is redeclared. " ++
          str "The new declaration (reflevity proved by " ++
           prterm th'.refl_refl ++
           (match th'.refl_sym with
               None -> str ""
             | Some t -> str " and symmetry proved by " ++ prterm t) ++
           str ") replaces the old declaration (reflexivity proved by "++
           prterm old_relation.refl_refl ++
           (match old_relation.refl_sym with
               None -> str ""
             | Some t -> str " and symmetry proved by " ++ prterm t) ++
           str ").") ;
        th'
     end
    else
     th
   in
    relation_table_add (s,th')
  and subst_set (_,subst,(s,th as obj)) =
    let s' = subst_mps subst s in
    let th' = subst_relation subst th in
      if s' == s && th' == th then obj else
        (s',th')
  and export_set x = Some x 
  in 
    declare_object {(default_object "relation-theory") with
                      cache_function = cache_set;
                      load_function = (fun i o -> cache_set o);
                      subst_function = subst_set;
                      classify_function = (fun (_,x) -> Substitute x);
                      export_function = export_set}

(******************************* Table of declared morphisms ********************)

(* Setoids are stored in a table which is synchronised with the Reset mechanism. *)

let morphism_table = ref Gmap.empty

let morphism_table_find m = Gmap.find m !morphism_table
let morphism_table_add (m,c) =
 let old =
  try
   morphism_table_find m
  with
   Not_found -> []
 in
  try
   let old_morph =
    List.find
     (function mor -> mor.args = c.args && mor.output = c.output) old
   in
    msg_warning
     (str "The morphism " ++ prmorphism m old_morph ++ str " is redeclared. " ++
      str "The new declaration whose compatibility is granted by " ++
       prterm c.lem ++ str " replaces the old declaration whose" ++
       str " compatibility was granted by " ++
       prterm old_morph.lem ++ str ".")
  with
   Not_found -> morphism_table := Gmap.add m (c::old) !morphism_table

let subst_morph subst morph = 
  let lem' = subst_mps subst morph.lem in
  let args' = list_smartmap (subst_mps_in_relation_class subst) morph.args in
  let output' = subst_mps_in_relation_class subst morph.output in
    if lem' == morph.lem
      && args' == morph.args
      && output' == morph.output
    then
      morph
    else
      { args = args' ;
        output = output' ;
        lem = lem'
      }


let _ = 
  Summary.declare_summary "morphism-table"
    { Summary.freeze_function = (fun () -> !morphism_table);
      Summary.unfreeze_function = (fun t -> morphism_table := t);
      Summary.init_function = (fun () -> morphism_table := Gmap .empty);
      Summary.survive_module = true;
      Summary.survive_section = false }

(* Declare a new type of object in the environment : "morphism-definition". *)

let (morphism_to_obj, obj_to_morphism)=
  let cache_set (_,(m, c)) = morphism_table_add (m, c)
  and subst_set (_,subst,(m,c as obj)) = 
    let m' = subst_mps subst m in
    let c' = subst_morph subst c in
      if m' == m && c' == c then obj else
        (m',c')
  and export_set x = Some x 
  in 
    declare_object {(default_object "morphism-definition") with
                      cache_function = cache_set;
                      load_function = (fun i o -> cache_set o);
                      subst_function = subst_set;
                      classify_function = (fun (_,x) -> Substitute x);
                      export_function = export_set}

(************************** Printing relations and morphisms **********************)

(*CSC: still "setoids" in the name *)
let print_setoids () =
  Gmap.iter
   (fun k relation ->
     assert (k=relation.refl_aeq) ;
     (*CSC: still "Setoid" in the error message *)
     ppnl (str"Setoid " ++ prrelation relation ++
      str"; reflexivity granted by " ++ prterm relation.refl_refl ++
      (match relation.refl_sym with
          None -> str ""
        | Some t -> str " symmetry granted by " ++ prterm t)))
   !relation_table ;
  Gmap.iter
   (fun k l ->
     List.iter
      (fun ({lem=lem} as mor) ->
        ppnl (str "Morphism " ++ prmorphism k mor ++
        str ". Compatibility granted by " ++
        prterm lem ++ str "."))
      l) !morphism_table
;;

(************************** Adding a relation to the database *********************)

let int_add_relation a aeq refl sym =
 match refl with
    None -> assert false (*CSC: unimplemented yet*)
  | Some refl ->
     let env = Global.env () in
       if
        (is_conv env Evd.empty (Typing.type_of env Evd.empty refl)
         (mkApp ((Lazy.force coq_is_reflexive), [| a; aeq |])))
       then
        Lib.add_anonymous_leaf 
         (relation_to_obj 
          (aeq, { refl_a = a;
                  refl_aeq = aeq;
                  refl_refl = refl;
                  refl_sym = sym}))
        else
         errorlabstrm "Add Relation Class"
          (str "Not a valid proof of reflexivity")

(* The vernac command "Add Relation ..." *)
let add_relation a aeq refl sym =
 int_add_relation (constr_of a) (constr_of aeq) (option_app constr_of refl)
  (option_app constr_of sym)

(***************** Adding a morphism to the database ****************************)

(* We maintain a table of the currently edited proofs of morphism lemma
   in order to add them in the morphism_table when the user does Save *)

let edited = ref Gmap.empty

let new_edited id m = 
  edited := Gmap.add id m !edited

let is_edited id =
  Gmap.mem id !edited

let no_more_edited id =
  edited := Gmap.remove id !edited

let what_edited id =
  Gmap.find id !edited

let check_is_dependent t n =
  let rec aux t i n =
    if (i<n)
    then (dependent (mkRel i) t) || (aux t (i+1) n)
    else false
  in aux t 0 n

(*CSC: I do not like this very much. I would prefer relation classes
  to be CIC objects. Keeping backward compatibility, however, is annoying. *)
let cic_relation_class_of_relation_class =
 function
    ACReflexive {refl_a = refl_a; refl_aeq = refl_aeq; refl_refl = refl_refl} ->
     mkApp ((Lazy.force coq_Reflexive), [| refl_a ; refl_aeq; refl_refl |])
  | ACLeibniz t -> mkApp ((Lazy.force coq_Leibniz), [| t |])

let gen_compat_lemma_statement output args m =
 let rec mk_signature =
  function
     [] -> assert false
   | [last] ->
      mkApp ((Lazy.force coq_singl), [| Lazy.force coq_Relation_Class; last |])
   | he::tl ->
      mkApp ((Lazy.force coq_cons),
       [| Lazy.force coq_Relation_Class; he ; mk_signature tl |])
 in
  let output = cic_relation_class_of_relation_class output in
  let args= mk_signature (List.map cic_relation_class_of_relation_class args) in
   args, output,
    mkApp ((Lazy.force coq_make_compatibility_goal), [| args ; output ; m |])

let new_morphism m id hook =
  let env = Global.env() in
  let typeofm = (Typing.type_of env Evd.empty m) in
  let typ = (nf_betaiota typeofm) in
  let (argsrev, output) = (decompose_prod typ) in
  let args_ty = (List.map snd (List.rev argsrev)) in
    if (args_ty=[])
    then errorlabstrm "New Morphism"
     (str "The term " ++ prterm m ++ str " is not a product")
    else if (check_is_dependent typ (List.length args_ty))
    then  errorlabstrm "New Morphism"
     (str "The term " ++ prterm m ++ str" should not be a dependent product")
    else
     begin
      let args = List.map default_relation_for_carrier args_ty in
      let output = default_relation_for_carrier output in
      let argsconstr,outputconstr,lem =
       gen_compat_lemma_statement output args m
      in
       new_edited id (m,args,argsconstr,output,outputconstr);
       Pfedit.start_proof id (IsGlobal (Proof Lemma)) 
        (Declare.clear_proofs (Global.named_context ()))
        lem hook;
       (* "unfold make_compatibility_goal" *)
       Pfedit.by
        (Tactics.unfold_constr
         (Lazy.force coq_make_compatibility_goal_ref)) ;
       (* "unfold make_compatibility_goal_aux" *)
       Pfedit.by
        (Tactics.unfold_constr
         (Lazy.force coq_make_compatibility_goal_aux_ref)) ;
       (* "simpl" *)
       Pfedit.by Tactics.simpl_in_concl ;
       Options.if_verbose msg (Pfedit.pr_open_subgoals ());
     end

let add_morphism lemma_infos mor_name (m,args,output) =
 let env = Global.env() in
 begin
  match lemma_infos with
     None -> () (* the Morphism_Theory object has alrady been created *)
   | Some (lem_name,argsconstr,outputconstr) ->
      (* only the compatibility has been proved; we need to declare the
         Morphism_Theory object *)
     let mext = current_constant lem_name in
     ignore (
      Declare.declare_constant mor_name
       (DefinitionEntry
         {const_entry_body =
           mkApp ((Lazy.force coq_Build_Morphism_Theory),
            [| argsconstr; outputconstr; m ; mext |]);
          const_entry_type = None;
          const_entry_opaque = false},
           IsDefinition))
  end ;
  let mmor = current_constant mor_name in
  let args_constr =
   List.map constr_relation_class_of_relation_relation_class args in
  let output_constr = constr_relation_class_of_relation_relation_class output in
   Lib.add_anonymous_leaf
    (morphism_to_obj (m, 
      { args = args_constr;
        output = output_constr;
        lem = mmor }));
   Options.if_verbose ppnl (prterm m ++ str " is registered as a morphism")   

let morphism_hook stre ref =
  let pf_id = id_of_global ref in
  let mor_id = id_of_string (string_of_id pf_id ^ "_morphism_theory") in
  let (m,args,argsconstr,output,outputconstr) = what_edited pf_id in
  if (is_edited pf_id)
  then 
    add_morphism (Some (pf_id,argsconstr,outputconstr)) mor_id
     (m,args,output); no_more_edited pf_id

let new_named_morphism id m = new_morphism (constr_of m) id morphism_hook

(************************ Add Setoid ******************************************)

(*CSC: I do not like this. I would prefer more self-describing constant names *)
let gen_morphism_name =
  let i = ref 0 in
    function () -> 
      incr i;
      make_ident "morphism_of_setoid_equality" (Some !i)

(* The vernac command "Add Setoid" *)
let add_setoid a aeq th =
 let a = constr_of a in
 let aeq = constr_of aeq in
 let th = constr_of th in
 let env = Global.env () in
  if is_conv env Evd.empty (Typing.type_of env Evd.empty th)
    (mkApp ((Lazy.force coq_Setoid_Theory), [| a; aeq |]))
  then
   begin
    let refl = mkApp ((Lazy.force coq_seq_refl), [| a; aeq; th |]) in
    let sym = mkApp ((Lazy.force coq_seq_sym), [| a; aeq; th |]) in
    int_add_relation a aeq (Some refl) (Some sym) ;
    let mor_name = gen_morphism_name () in
    let _ =
     Declare.declare_constant mor_name
      (DefinitionEntry
        {const_entry_body =
          mkApp
           ((Lazy.force coq_equality_morphism_of_setoid_theory),
            [| a; aeq; th |]) ;
         const_entry_type = None;
         const_entry_opaque = false},
        IsDefinition) in
    let aeq_rel =
     ACReflexive
      {refl_a = a ; refl_aeq = aeq ; refl_refl = refl ; refl_sym = (Some sym)}
    in
     add_morphism None mor_name
      (aeq,[aeq_rel;aeq_rel],Lazy.force coq_prop_relation)
   end
  else
   errorlabstrm "Add Setoid" (str "Not a valid setoid theory")


(****************************** The tactic itself *******************************)

type constr_with_marks =
  | MApp of constr * morphism_class * constr_with_marks array 
  | Toreplace
  | Tokeep of constr
  | Mimp of constr_with_marks * constr_with_marks

let is_to_replace = function
 | Tokeep _ -> false
 | Toreplace -> true
 | MApp _ -> true
 | Mimp _ -> true

let get_mark a = 
  Array.fold_left (||) false (Array.map is_to_replace a)

type argument =
   Toapply of constr         (* apply the function to the argument *)
 | Toexpand of name * types  (* beta-expand the function w.r.t. an argument
                                of this type *)
let beta_expand c args_rev =
 let rec to_expand =
  function
     [] -> []
   | (Toapply _)::tl -> to_expand tl
   | (Toexpand (name,s))::tl -> (name,s)::(to_expand tl) in
 let rec aux n =
  function
     [] -> []
   | (Toapply arg)::tl -> arg::(aux n tl)
   | (Toexpand _)::tl -> (mkRel n)::(aux (n + 1) tl)
 in
  compose_lam (to_expand args_rev)
   (mkApp (c, Array.of_list (List.rev (aux 1 args_rev))))

let relation_of_hypothesis_and_term_to_rewrite gl (_,h) t =
 let hypt = pf_type_of gl h in
 let (heq, hargs) = decompose_app hypt in
 let rec get_all_but_last_two =
  function
     []
   | [_] -> assert false
   | [_;_] -> []
   | he::tl -> he::(get_all_but_last_two tl) in
 let aeq = mkApp (heq,(Array.of_list (get_all_but_last_two hargs))) in
  try
   relation_table_find aeq
  with Not_found ->
   (*CSC: still "setoid" in the error message *)
   errorlabstrm "Setoid_rewrite"
    (prterm aeq ++ str " is not a setoid equality.")

let mark_occur t in_c =
 let rec aux t in_c =
  if eq_constr t in_c then
    Toreplace
  else
    match kind_of_term in_c with
      | App (c,al) -> 
          let a = Array.map (aux t) al in
           if not (get_mark a) then Tokeep in_c
           else
            let mor =
              try
                begin
                 match morphism_table_find c with
                    [] -> assert false
                  | morphism::tl ->
                     if tl <> [] then
                      msg_warning
                       (str "There are several morphisms declared for " ++
                         prterm c ++
                         str ". The morphism " ++ prmorphism c morphism ++
                         str " is randomly chosen.") ;
                     Some
                      (ACMorphism
                       (relation_morphism_of_constr_morphism morphism))
                end
               with Not_found -> None
            in
             (match mor with
                 Some mor -> MApp (c,mor,a)
               | None ->
                 (* the tactic works only if the function type is
                    made of non-dependent products only. However, here we
                    can cheat a bit by partially istantiating c to match
                    the requirement when the arguments to be replaced are
                    bound by non-dependent products only. *)
                 let env = Global.env() in
                 let typeofc = (Typing.type_of env Evd.empty c) in
                 let typ = nf_betaiota typeofc in
                 let rec find_non_dependent_function env c c_args_rev typ
                          f_args_rev a_rev=
                  function
                     [] ->
                      let mor =
                       ACFunction {f_args=List.rev f_args_rev ; f_output=typ} in
                      let func = beta_expand c c_args_rev in
                       MApp (func,mor,Array.of_list (List.rev a_rev))
                   | (he::tl) as a->
                      let typnf = Reduction.whd_betadeltaiota env typ in
                       match kind_of_term typnf with
                         Cast (typ,_) ->
                          find_non_dependent_function env c c_args_rev typ
                           f_args_rev a_rev a
                       | Prod (name,s,t) ->
                          let env' = push_rel (name,None,s) env in
                          begin
                           match noccurn 1 t, he with
                              _, Tokeep arg ->
                               (* generic product, to keep *)
                               find_non_dependent_function
                                env' c ((Toapply arg)::c_args_rev)
                                (subst1 arg t) f_args_rev a_rev tl
                            | true, _ ->
                               (* non-dependent product, to replace *)
                               find_non_dependent_function
                                env' c ((Toexpand (name,s))::c_args_rev)
                                 (lift 1 t) (s::f_args_rev) (he::a_rev) tl
                            | false, _ ->
                               (* dependent product, to replace *)
                               (*CSC: this limitation is due to the reflexive
                                 implementation and it is hard to lift *)
                               errorlabstrm "Setoid_replace"
                                (str "Cannot rewrite in the argument of a " ++
                                 str "dependent product. If you need this " ++
                                 str "feature, please report to the authors.")
                          end
                       | _ -> assert false
                 in
                  find_non_dependent_function env c [] typ [] []
                   (Array.to_list a))
      | Prod (_, c1, c2) -> 
          if (dependent (mkRel 1) c2)
          then Tokeep in_c
          else 
            let c1m = aux t c1 in
            let c2m = aux t c2 in
              if ((is_to_replace c1m)||(is_to_replace c2m))
              then (Mimp (c1m, c2m))
              else Tokeep in_c
      | _ -> Tokeep in_c
 in aux t in_c

let cic_morphism_context_list_of_list hole_relation =
 let rec aux =
  function
     [] -> assert false
   | [out,value] ->
      mkApp ((Lazy.force coq_singl), [| Lazy.force coq_Relation_Class ; out |]),
      mkApp ((Lazy.force coq_fcl_singl), [| hole_relation; out ; value |])
   | (out,value)::tl ->
      let outtl, valuetl = aux tl in
       mkApp ((Lazy.force coq_cons),
        [| Lazy.force coq_Relation_Class ; out ; outtl |]),
       mkApp ((Lazy.force coq_fcl_cons),
        [| hole_relation ; out ; outtl ; value ; valuetl |])
 in aux

let rec cic_type_nelist_of_list =
 function
    [] -> assert false
  | [value] ->
      mkApp ((Lazy.force coq_singl), [| mkType (Termops.new_univ ()) ; value |])
  | value::tl ->
     mkApp ((Lazy.force coq_cons),
      [| mkType (Termops.new_univ ()); value; cic_type_nelist_of_list tl |])

let syntactic_but_representation_of_marked_but hole_relation =
 let rec aux out =
  function
     MApp (f, m, args) ->
      let morphism_theory, relations =
       match m with
          ACMorphism { args = args ; lem = lem } -> lem,args
        | ACFunction { f_args = f_args ; f_output = f_output } ->
           let mt =
            if eq_constr out (cic_relation_class_of_relation_class
              (Lazy.force coq_prop_relation))
            then
              mkApp ((Lazy.force coq_morphism_theory_of_predicate),
               [| cic_type_nelist_of_list f_args; f|])
            else
              mkApp ((Lazy.force coq_morphism_theory_of_function),
               [| cic_type_nelist_of_list f_args; f_output; f|])
           in
            mt,List.map (fun x -> ACLeibniz x) f_args in
      let cic_relations =
       List.map cic_relation_class_of_relation_class relations in
      let cic_args_relations,argst =
       cic_morphism_context_list_of_list hole_relation
        (List.combine cic_relations
          (List.map2 (fun t v -> aux t v) cic_relations
           (Array.to_list args)))
      in
       mkApp ((Lazy.force coq_App),
        [|hole_relation ; cic_args_relations ; out ; morphism_theory ; argst|])
   | Toreplace ->
      mkApp ((Lazy.force coq_Toreplace), [| hole_relation |])
   | Tokeep c ->
       mkApp ((Lazy.force coq_Tokeep), [| hole_relation ; out ; c |])
   | Mimp (source, target) ->
      mkApp ((Lazy.force coq_Imp),
       [| hole_relation ; aux out source ; aux out target |])
 in aux

let apply_coq_setoid_rewrite hole_relation c1 c2 (lft2rgt,h) m =
 let {refl_aeq=hole_equality; refl_sym=hole_symmetry} = hole_relation in
 let hole_relation =
  cic_relation_class_of_relation_class (ACReflexive hole_relation) in
 let prop_relation =
  cic_relation_class_of_relation_class (Lazy.force coq_prop_relation) in
 let hyp =
  if lft2rgt then h else
   match hole_symmetry with
      Some sym ->
       mkApp (sym, [| c2 ; c1 ; h |])
    | None ->
       errorlabstrm "Setoid_rewrite"
        (str "Rewriting from right to left not permitted since the " ++
         str "relation " ++ prterm hole_equality ++ str " is not known " ++
         str "to be symmetric. Either declare it as a symmetric " ++
         str "relation or use setoid_replace or (setoid_)rewrite from " ++
         str "left to right only.")
 in
  mkApp ((Lazy.force coq_setoid_rewrite),
   [| hole_relation ; prop_relation ; c1 ; c2 ;
      syntactic_but_representation_of_marked_but hole_relation
       prop_relation m ; hyp |])

let relation_rewrite c1 c2 (lft2rgt,cl) gl =
 let but = pf_concl gl in
 let (hyp,c1,c2) =
  let (cl',_) = Clenv.unify_to_subterm cl (c1,but) in
  let c1 = Clenv.clenv_instance_term cl' c1 in
  let c2 = Clenv.clenv_instance_term cl' c2 in
   (lft2rgt,Clenv.clenv_instance_template cl'), c1, c2
 in
  let input_relation = relation_of_hypothesis_and_term_to_rewrite gl hyp c1 in
  let marked_but = mark_occur c1 but in
  let th =
   apply_coq_setoid_rewrite input_relation c1 c2 hyp marked_but
  in
   let thty = pf_type_of gl th in
   let simplified_thty =
    pf_unfoldn [[], Lazy.force coq_iff] gl
     (pf_unfoldn [[], Lazy.force coq_relation_of_relation_class] gl thty)
   in
    let ty1,ty2 =
     match destApplication simplified_thty with
        (_ (*and*),[|ty1;ty2|]) -> ty1,ty2
      | _ -> assert false
    in
     let th' = mkApp ((Lazy.force coq_proj2), [|ty1; ty2; th|]) in
     let hyp2 = Termops.replace_term c1 c2 but in
     let simplified_thty' = mkProd (Anonymous, hyp2, lift 1  but) in
      (*CSC: the next line does not work because simplified_thty' is not
        used to generate the type of the new goals ;-(
      Tactics.apply (mkCast (th',simplified_thty')) gl
        Thus I am using the following longer (and less accurate) code *)
       tclTHENS
        (Tactics.apply (mkCast (th',simplified_thty')))
        [Tactics.change_in_concl None hyp2] gl
     (*CSC: this is another way to proceed, but the generated proof_term
       would be much bigger than possible
      Tactics.forward true Anonymous (mkCast (th',simplified_thty')) gl
      (... followed by the application of the introduced hypothesis ...)
     *)

let general_s_rewrite lft2rgt c gl =
 let (wc,_)  = Evar_refiner.startWalk gl in
 let ctype = pf_type_of gl c in
 let eqclause  =
  Clenv.make_clenv_binding wc (c,ctype) Rawterm.NoBindings in
 let (equiv, args) =
  decompose_app (Clenv.clenv_instance_template_type eqclause) in
 let rec get_last_two = function
   | [c1;c2] -> (c1, c2)
   | x::y::z -> get_last_two (y::z)
   | _ -> error "The term provided is not an equivalence" in
 let (c1,c2) = get_last_two args in
   if lft2rgt
   then relation_rewrite c1 c2 (lft2rgt,eqclause) gl
   else relation_rewrite c2 c1 (lft2rgt,eqclause) gl

exception Use_replace

(*CSC: the name should be changed *)
let setoid_replace c1 c2 gl =
 try
  let relation =
   match default_relation_for_carrier (pf_type_of gl c1) with
      ACReflexive sa -> sa
    | ACLeibniz _ -> raise Use_replace
  in
   let eq = mkApp (relation.refl_aeq, [| c1 ; c2 |]) in
    tclTHENS (assert_tac false Anonymous eq)
      [onLastHyp (fun id ->
        tclTHEN
          (general_s_rewrite true (mkVar id))
          (clear [id]));
       Tacticals.tclIDTAC] gl
 with
  Use_replace -> (!replace c1 c2) gl

let setoid_rewriteLR = general_s_rewrite true

let setoid_rewriteRL = general_s_rewrite false