(* Copyright (C) 2005, HELM Team.
 * 
 * This file is part of HELM, an Hypertextual, Electronic
 * Library of Mathematics, developed at the Computer Science
 * Department, University of Bologna, Italy.
 * 
 * HELM is free software; you can redistribute it and/or
 * modify it under the terms of the GNU General Public License
 * as published by the Free Software Foundation; either version 2
 * of the License, or (at your option) any later version.
 * 
 * HELM is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with HELM; if not, write to the Free Software
 * Foundation, Inc., 59 Temple Place - Suite 330, Boston,
 * MA  02111-1307, USA.
 * 
 * For details, see the HELM World-Wide-Web page,
 * http://cs.unibo.it/helm/.
 *)

let debug_print = Utils.debug_print;;


type retrieval_mode = Matching | Unification;;


let print_candidates mode term res =
  let _ =
    match mode with
    | Matching ->
        Printf.printf "| candidates Matching %s\n" (CicPp.ppterm term)
    | Unification ->
        Printf.printf "| candidates Unification %s\n" (CicPp.ppterm term)
  in
  print_endline
    (String.concat "\n"
       (List.map
          (fun (p, e) ->
             Printf.sprintf "| (%s, %s)" (Utils.string_of_pos p)
               (Inference.string_of_equality e))
          res));
  print_endline "|";
;;


let indexing_retrieval_time = ref 0.;;


(* let my_apply_subst subst term = *)
(*   let module C = Cic in *)
(*   let lookup lift_amount meta = *)
(*     match meta with *)
(*     | C.Meta (i, _) -> ( *)
(*         try *)
(*           let _, (_, t, _) = List.find (fun (m, _) -> m = i) subst in *)
(*           (\* CicSubstitution.lift lift_amount  *\)t *)
(*         with Not_found -> meta *)
(*       ) *)
(*     | _ -> assert false *)
(*   in *)
(*   let rec apply_aux lift_amount =  function *)
(*     | C.Meta (i, l) as t -> lookup lift_amount t *)
(*     | C.Appl l -> C.Appl (List.map (apply_aux lift_amount) l) *)
(*     | C.Prod (nn, s, t) -> *)
(*         C.Prod (nn, apply_aux lift_amount s, apply_aux (lift_amount+1) t) *)
(*     | C.Lambda (nn, s, t) -> *)
(*         C.Lambda (nn, apply_aux lift_amount s, apply_aux (lift_amount+1) t) *)
(*     | t -> t *)
(*   in *)
(*   apply_aux 0 term *)
(* ;; *)


(* let apply_subst subst term = *)
(*   Printf.printf "| apply_subst:\n| subst: %s\n| term: %s\n" *)
(*     (Utils.print_subst ~prefix:" ; " subst) (CicPp.ppterm term); *)
(*   let res = my_apply_subst subst term in *)
(* (\*   let res = CicMetaSubst.apply_subst subst term in *\) *)
(*   Printf.printf "| res: %s\n" (CicPp.ppterm res); *)
(*   print_endline "|"; *)
(*   res *)
(* ;; *)

(* let apply_subst = my_apply_subst *)
let apply_subst = CicMetaSubst.apply_subst


(* let apply_subst = *)
(*   let profile = CicUtil.profile "apply_subst" in *)
(*   (fun s a -> profile (apply_subst s) a) *)
(* ;; *)


(*
(* NO INDEXING *)
let empty_table () = []

let index table equality =
  let _, _, (_, l, r, ordering), _, _ = equality in
  match ordering with
  | Utils.Gt -> (Utils.Left, equality)::table
  | Utils.Lt -> (Utils.Right, equality)::table
  | _ -> (Utils.Left, equality)::(Utils.Right, equality)::table
;;

let remove_index table equality =
  List.filter (fun (p, e) -> e != equality) table
;;

let in_index table equality =
  List.exists (fun (p, e) -> e == equality) table
;;

let get_candidates mode table term = table
*)


(*
(* PATH INDEXING *)
let empty_table () =
  Path_indexing.PSTrie.empty
;;

let index = Path_indexing.index
and remove_index = Path_indexing.remove_index
and in_index = Path_indexing.in_index;;
  
let get_candidates mode trie term =
  let t1 = Unix.gettimeofday () in
  let res = 
    let s = 
      match mode with
      | Matching -> Path_indexing.retrieve_generalizations trie term
      | Unification -> Path_indexing.retrieve_unifiables trie term
(*       Path_indexing.retrieve_all trie term *)
    in
    Path_indexing.PosEqSet.elements s
  in
(*   print_candidates mode term res; *)
  let t2 = Unix.gettimeofday () in
  indexing_retrieval_time := !indexing_retrieval_time +. (t2 -. t1);
  res
;;
*)


(* DISCRIMINATION TREES *)
let init_index () =
  Hashtbl.clear Discrimination_tree.arities;
;;

let empty_table () =
  Discrimination_tree.DiscriminationTree.empty
;;

let index = Discrimination_tree.index
and remove_index = Discrimination_tree.remove_index
and in_index = Discrimination_tree.in_index;;

let get_candidates mode tree term =
  let t1 = Unix.gettimeofday () in
  let res =
    let s = 
      match mode with
      | Matching -> Discrimination_tree.retrieve_generalizations tree term
      | Unification -> Discrimination_tree.retrieve_unifiables tree term
    in
    Discrimination_tree.PosEqSet.elements s
  in
(*   print_candidates mode term res; *)
(*   print_endline (Discrimination_tree.string_of_discrimination_tree tree); *)
(*   print_newline (); *)
  let t2 = Unix.gettimeofday () in
  indexing_retrieval_time := !indexing_retrieval_time +. (t2 -. t1);
  res
;;


(* let get_candidates = *)
(*   let profile = CicUtil.profile "Indexing.get_candidates" in *)
(*   (fun mode tree term -> profile.profile (get_candidates mode tree) term) *)
(* ;; *)


let match_unif_time_ok = ref 0.;;
let match_unif_time_no = ref 0.;;


let rec find_matches metasenv context ugraph lift_amount term termty =
  let module C = Cic in
  let module U = Utils in
  let module S = CicSubstitution in
  let module M = CicMetaSubst in
  let module HL = HelmLibraryObjects in
  let cmp = !Utils.compare_terms in
(*   let names = Utils.names_of_context context in *)
(*   let termty, ugraph = *)
(*     CicTypeChecker.type_of_aux' metasenv context term ugraph *)
(*   in *)
  function
    | [] -> None
    | candidate::tl ->
        let pos, (_, proof, (ty, left, right, o), metas, args) = candidate in
(*         if not (fst (CicReduction.are_convertible *)
(*                        ~metasenv context termty ty ugraph)) then ( *)
(* (\*           debug_print (lazy ( *\) *)
(* (\*             Printf.sprintf "CANDIDATE HAS WRONG TYPE: %s required, %s found" *\) *)
(* (\*               (CicPp.pp termty names) (CicPp.pp ty names))); *\) *)
(*           find_matches metasenv context ugraph lift_amount term termty tl *)
(*         ) else *)
          let do_match c (* other *) eq_URI =
            let subst', metasenv', ugraph' =
              let t1 = Unix.gettimeofday () in
              try
                let r =
                  Inference.matching (metasenv @ metas) context
                    term (S.lift lift_amount c) ugraph in
                let t2 = Unix.gettimeofday () in
                match_unif_time_ok := !match_unif_time_ok +. (t2 -. t1);
                r
              with Inference.MatchingFailure as e ->
                let t2 = Unix.gettimeofday () in
                match_unif_time_no := !match_unif_time_no +. (t2 -. t1);
                raise e
            in
            Some (C.Rel (1 + lift_amount), subst', metasenv', ugraph',
                  (candidate, eq_URI))
          in
          let c, other, eq_URI =
            if pos = Utils.Left then left, right, Utils.eq_ind_URI ()
            else right, left, Utils.eq_ind_r_URI ()
          in
          if o <> U.Incomparable then
            try
              do_match c (* other *) eq_URI
            with Inference.MatchingFailure ->
              find_matches metasenv context ugraph lift_amount term termty tl
          else
            let res =
              try do_match c (* other *) eq_URI
              with Inference.MatchingFailure -> None
            in
            match res with
            | Some (_, s, _, _, _) ->
                let c' = (* M. *)apply_subst s c
                and other' = (* M. *)apply_subst s other in
                let order = cmp c' other' in
                let names = U.names_of_context context in
(*                 let _ = *)
(*                   debug_print *)
(*                     (Printf.sprintf "OK matching: %s and %s, order: %s" *)
(*                        (CicPp.ppterm c') *)
(*                        (CicPp.ppterm other') *)
(*                        (Utils.string_of_comparison order)); *)
(*                   debug_print *)
(*                     (Printf.sprintf "subst:\n%s\n" (Utils.print_subst s)) *)
(*                 in *)
                if order = U.Gt then
                  res
                else
                  find_matches
                    metasenv context ugraph lift_amount term termty tl
            | None ->
                find_matches metasenv context ugraph lift_amount term termty tl
;;


let rec find_all_matches ?(unif_fun=Inference.unification)
    metasenv context ugraph lift_amount term termty =
  let module C = Cic in
  let module U = Utils in
  let module S = CicSubstitution in
  let module M = CicMetaSubst in
  let module HL = HelmLibraryObjects in
  let cmp = !Utils.compare_terms in
(*   let names = Utils.names_of_context context in *)
(*   let termty, ugraph = *)
(*     CicTypeChecker.type_of_aux' metasenv context term ugraph *)
(*   in *)
(*   let _ = *)
(*     match term with *)
(*     | C.Meta _ -> assert false *)
(*     | _ -> () *)
(*   in *)
  function
    | [] -> []
    | candidate::tl ->
        let pos, (_, _, (ty, left, right, o), metas, args) = candidate in
(*         if not (fst (CicReduction.are_convertible *)
(*                        ~metasenv context termty ty ugraph)) then ( *)
(* (\*           debug_print (lazy ( *\) *)
(* (\*             Printf.sprintf "CANDIDATE HAS WRONG TYPE: %s required, %s found" *\) *)
(* (\*               (CicPp.pp termty names) (CicPp.pp ty names))); *\) *)
(*           find_all_matches ~unif_fun metasenv context ugraph *)
(*             lift_amount term termty tl *)
(*         ) else *)
          let do_match c (* other *) eq_URI =
            let subst', metasenv', ugraph' =
              let t1 = Unix.gettimeofday () in
              try
                let r = 
                  unif_fun (metasenv @ metas) context
                    term (S.lift lift_amount c) ugraph in
                let t2 = Unix.gettimeofday () in
                match_unif_time_ok := !match_unif_time_ok +. (t2 -. t1);
                r
              with
              | Inference.MatchingFailure
              | CicUnification.UnificationFailure _
              | CicUnification.Uncertain _ as e ->
                let t2 = Unix.gettimeofday () in
                match_unif_time_no := !match_unif_time_no +. (t2 -. t1);
                raise e
            in
            (C.Rel (1 + lift_amount), subst', metasenv', ugraph',
             (candidate, eq_URI))
          in
          let c, other, eq_URI =
            if pos = Utils.Left then left, right, Utils.eq_ind_URI ()
            else right, left, Utils.eq_ind_r_URI ()
          in
          if o <> U.Incomparable then
            try
              let res = do_match c (* other *) eq_URI in
              res::(find_all_matches ~unif_fun metasenv context ugraph
                      lift_amount term termty tl)
            with
            | Inference.MatchingFailure
            | CicUnification.UnificationFailure _
            | CicUnification.Uncertain _ ->
              find_all_matches ~unif_fun metasenv context ugraph
                lift_amount term termty tl
          else
            try
              let res = do_match c (* other *) eq_URI in
              match res with
              | _, s, _, _, _ ->
                  let c' = (* M. *)apply_subst s c
                  and other' = (* M. *)apply_subst s other in
                  let order = cmp c' other' in
                  let names = U.names_of_context context in
                  if order <> U.Lt && order <> U.Le then
                    res::(find_all_matches ~unif_fun metasenv context ugraph
                            lift_amount term termty tl)
                  else
                    find_all_matches ~unif_fun metasenv context ugraph
                      lift_amount term termty tl
            with
            | Inference.MatchingFailure
            | CicUnification.UnificationFailure _
            | CicUnification.Uncertain _ ->
                find_all_matches ~unif_fun metasenv context ugraph
                  lift_amount term termty tl
;;


let subsumption env table target =
  let _, (ty, left, right, _), tmetas, _ = target in
  let metasenv, context, ugraph = env in
  let metasenv = metasenv @ tmetas in
  let samesubst subst subst' =
    let tbl = Hashtbl.create (List.length subst) in
    List.iter (fun (m, (c, t1, t2)) -> Hashtbl.add tbl m (c, t1, t2)) subst;
    List.for_all
      (fun (m, (c, t1, t2)) ->
         try
           let c', t1', t2' = Hashtbl.find tbl m in
           if (c = c') && (t1 = t1') && (t2 = t2') then true
           else false
         with Not_found ->
           true)
      subst'
  in
  let leftr =
    match left with
    | Cic.Meta _ -> []
    | _ ->
        let leftc = get_candidates Matching table left in
        find_all_matches ~unif_fun:Inference.matching
          metasenv context ugraph 0 left ty leftc
  in
  let rec ok what = function
    | [] -> false, []
    | (_, subst, menv, ug, ((pos, (_, _, (_, l, r, o), _, _)), _))::tl ->
        try
          let other = if pos = Utils.Left then r else l in
          let subst', menv', ug' =
            let t1 = Unix.gettimeofday () in
            try
              let r = 
                Inference.matching metasenv context what other ugraph in
              let t2 = Unix.gettimeofday () in
              match_unif_time_ok := !match_unif_time_ok +. (t2 -. t1);
              r
            with Inference.MatchingFailure as e ->
              let t2 = Unix.gettimeofday () in
              match_unif_time_no := !match_unif_time_no +. (t2 -. t1);
              raise e
          in
          if samesubst subst subst' then
            true, subst
          else
            ok what tl
        with Inference.MatchingFailure ->
          ok what tl
  in
  let r, subst = ok right leftr in
  if r then
    true, subst
  else
    let rightr =
      match right with
      | Cic.Meta _ -> []
      | _ ->
          let rightc = get_candidates Matching table right in
          find_all_matches ~unif_fun:Inference.matching
            metasenv context ugraph 0 right ty rightc
    in
    ok left rightr
;;


let rec demodulation_aux metasenv context ugraph table lift_amount term =
  let module C = Cic in
  let module S = CicSubstitution in
  let module M = CicMetaSubst in
  let module HL = HelmLibraryObjects in
  let candidates = get_candidates Matching table term in
  match term with
  | C.Meta _ -> None
  | term ->
      let termty, ugraph =
        C.Implicit None, ugraph
(*         CicTypeChecker.type_of_aux' metasenv context term ugraph *)
      in
      let res =
        find_matches metasenv context ugraph lift_amount term termty candidates
      in
      if res <> None then
        res
      else
        match term with
        | C.Appl l ->
            let res, ll = 
              List.fold_left
                (fun (res, tl) t ->
                   if res <> None then
                     (res, tl @ [S.lift 1 t])
                   else 
                     let r =
                       demodulation_aux metasenv context ugraph table
                         lift_amount t
                     in
                     match r with
                     | None -> (None, tl @ [S.lift 1 t])
                     | Some (rel, _, _, _, _) -> (r, tl @ [rel]))
                (None, []) l
            in (
              match res with
              | None -> None
              | Some (_, subst, menv, ug, eq_found) ->
                  Some (C.Appl ll, subst, menv, ug, eq_found)
            )
        | C.Prod (nn, s, t) ->
            let r1 =
              demodulation_aux metasenv context ugraph table lift_amount s in (
              match r1 with
              | None ->
                  let r2 =
                    demodulation_aux metasenv
                      ((Some (nn, C.Decl s))::context) ugraph
                      table (lift_amount+1) t
                  in (
                    match r2 with
                    | None -> None
                    | Some (t', subst, menv, ug, eq_found) ->
                        Some (C.Prod (nn, (S.lift 1 s), t'),
                              subst, menv, ug, eq_found)
                  )
              | Some (s', subst, menv, ug, eq_found) ->
                  Some (C.Prod (nn, s', (S.lift 1 t)),
                        subst, menv, ug, eq_found)
            )
        | C.Lambda (nn, s, t) ->
            let r1 =
              demodulation_aux metasenv context ugraph table lift_amount s in (
              match r1 with
              | None ->
                  let r2 =
                    demodulation_aux metasenv
                      ((Some (nn, C.Decl s))::context) ugraph
                      table (lift_amount+1) t
                  in (
                    match r2 with
                    | None -> None
                    | Some (t', subst, menv, ug, eq_found) ->
                        Some (C.Lambda (nn, (S.lift 1 s), t'),
                              subst, menv, ug, eq_found)
                  )
              | Some (s', subst, menv, ug, eq_found) ->
                  Some (C.Lambda (nn, s', (S.lift 1 t)),
                        subst, menv, ug, eq_found)
            )
        | t ->
            None
;;


let build_newtarget_time = ref 0.;;


let demod_counter = ref 1;;

let rec demodulation_equality newmeta env table sign target =
  let module C = Cic in
  let module S = CicSubstitution in
  let module M = CicMetaSubst in
  let module HL = HelmLibraryObjects in
  let metasenv, context, ugraph = env in
  let _, proof, (eq_ty, left, right, order), metas, args = target in
  let metasenv' = metasenv @ metas in

  let maxmeta = ref newmeta in
  
  let build_newtarget is_left (t, subst, menv, ug, (eq_found, eq_URI)) =
    let time1 = Unix.gettimeofday () in
    
    let pos, (_, proof', (ty, what, other, _), menv', args') = eq_found in
    let ty =
      try fst (CicTypeChecker.type_of_aux' metasenv context what ugraph)
      with CicUtil.Meta_not_found _ -> ty
    in
    let what, other = if pos = Utils.Left then what, other else other, what in
    let newterm, newproof =
      let bo = (* M. *)apply_subst subst (S.subst other t) in
(*       let t' = *)
(*         let name = C.Name ("x_Demod_" ^ (string_of_int !demod_counter)) in *)
(*         incr demod_counter; *)
(*         let l, r = *)
(*           if is_left then t, S.lift 1 right else S.lift 1 left, t in *)
(*         (name, ty, S.lift 1 eq_ty, l, r) *)
(*       in *)
      let name = C.Name ("x_Demod_" ^ (string_of_int !demod_counter)) in
      incr demod_counter;
      let bo' =
        let l, r = if is_left then t, S.lift 1 right else S.lift 1 left, t in
        C.Appl [C.MutInd (LibraryObjects.eq_URI (), 0, []);
                S.lift 1 eq_ty; l; r]
      in
      if sign = Utils.Positive then
        (bo,
         Inference.ProofBlock (
           subst, eq_URI, (name, ty), bo'(* t' *), eq_found, proof))
      else
        let metaproof = 
          incr maxmeta;
          let irl =
            CicMkImplicit.identity_relocation_list_for_metavariable context in
          Printf.printf "\nADDING META: %d\n" !maxmeta;
          print_newline ();
          C.Meta (!maxmeta, irl)
        in
(*         let target' = *)
          let eq_found =
            let proof' =
(*               let ens = *)
(*                 if pos = Utils.Left then *)
(*                   build_ens_for_sym_eq ty what other *)
(*                 else *)
(*                   build_ens_for_sym_eq ty other what *)
              let termlist =
                if pos = Utils.Left then [ty; what; other]
                else [ty; other; what]
              in
              Inference.ProofSymBlock (termlist, proof')
            in
            let what, other =
              if pos = Utils.Left then what, other else other, what
            in
            pos, (0, proof', (ty, other, what, Utils.Incomparable),
                  menv', args')
          in
          let target_proof =
            let pb =
              Inference.ProofBlock (subst, eq_URI, (name, ty), bo'(* t' *),
                                    eq_found, Inference.BasicProof metaproof)
            in
            match proof with
            | Inference.BasicProof _ ->
                print_endline "replacing a BasicProof";
                pb
            | Inference.ProofGoalBlock (_, parent_proof(* parent_eq *)) ->
                print_endline "replacing another ProofGoalBlock";
                Inference.ProofGoalBlock (pb, parent_proof(* parent_eq *))
            | _ -> assert false
          in
(*           (0, target_proof, (eq_ty, left, right, order), metas, args) *)
(*         in *)
        let refl =
          C.Appl [C.MutConstruct (* reflexivity *)
                    (LibraryObjects.eq_URI (), 0, 1, []);
                  eq_ty; if is_left then right else left]          
        in
        (bo,
         Inference.ProofGoalBlock (Inference.BasicProof refl, target_proof(* target' *)))
    in
    let left, right = if is_left then newterm, right else left, newterm in
    let m = (Inference.metas_of_term left) @ (Inference.metas_of_term right) in
    let newmetasenv = List.filter (fun (i, _, _) -> List.mem i m) metas
    and newargs = args
(*       let a =  *)
(*         List.filter *)
(*           (function C.Meta (i, _) -> List.mem i m | _ -> assert false) args in *)
(*       let delta = (List.length args) - (List.length a) in *)
(*       if delta > 0 then *)
(*         let first = List.hd a in *)
(*         let rec aux l = function *)
(*           | 0 -> l *)
(*           | d -> let l = aux l (d-1) in l @ [first] *)
(*         in *)
(*         aux a delta *)
(*       else *)
(*         a *)
    in
    let ordering = !Utils.compare_terms left right in

    let time2 = Unix.gettimeofday () in
    build_newtarget_time := !build_newtarget_time +. (time2 -. time1);

    let res =
      let w = Utils.compute_equality_weight eq_ty left right in
      (w, newproof, (eq_ty, left, right, ordering), newmetasenv, newargs)
    in
    !maxmeta, res
  in
  let res = demodulation_aux metasenv' context ugraph table 0 left in
  match res with
  | Some t ->
      let newmeta, newtarget = build_newtarget true t in
      if (Inference.is_identity (metasenv', context, ugraph) newtarget) ||
        (Inference.meta_convertibility_eq target newtarget) then
          newmeta, newtarget
      else
(*         if subsumption env table newtarget then *)
(*           newmeta, build_identity newtarget *)
(*         else *)
        demodulation_equality newmeta env table sign newtarget
  | None ->
      let res = demodulation_aux metasenv' context ugraph table 0 right in
      match res with
      | Some t ->
          let newmeta, newtarget = build_newtarget false t in
          if (Inference.is_identity (metasenv', context, ugraph) newtarget) ||
            (Inference.meta_convertibility_eq target newtarget) then
              newmeta, newtarget
          else
(*             if subsumption env table newtarget then *)
(*               newmeta, build_identity newtarget *)
(*             else *)
              demodulation_equality newmeta env table sign newtarget
      | None ->
          newmeta, target
;;


let rec betaexpand_term metasenv context ugraph table lift_amount term =
  let module C = Cic in
  let module S = CicSubstitution in
  let module M = CicMetaSubst in
  let module HL = HelmLibraryObjects in
  let candidates = get_candidates Unification table term in
  let res, lifted_term = 
    match term with
    | C.Meta (i, l) ->
        let l', lifted_l =
          List.fold_right
            (fun arg (res, lifted_tl) ->
               match arg with
               | Some arg ->
                   let arg_res, lifted_arg =
                     betaexpand_term metasenv context ugraph table
                       lift_amount arg in
                   let l1 =
                     List.map
                       (fun (t, s, m, ug, eq_found) ->
                          (Some t)::lifted_tl, s, m, ug, eq_found)
                       arg_res
                   in
                   (l1 @
                      (List.map
                         (fun (l, s, m, ug, eq_found) ->
                            (Some lifted_arg)::l, s, m, ug, eq_found)
                         res),
                    (Some lifted_arg)::lifted_tl)
               | None ->
                   (List.map
                      (fun (r, s, m, ug, eq_found) ->
                         None::r, s, m, ug, eq_found) res,
                    None::lifted_tl)
            ) l ([], [])
        in
        let e =
          List.map
            (fun (l, s, m, ug, eq_found) ->
               (C.Meta (i, l), s, m, ug, eq_found)) l'
        in
        e, C.Meta (i, lifted_l)
          
    | C.Rel m ->
        [], if m <= lift_amount then C.Rel m else C.Rel (m+1)
          
    | C.Prod (nn, s, t) ->
        let l1, lifted_s =
          betaexpand_term metasenv context ugraph table lift_amount s in
        let l2, lifted_t =
          betaexpand_term metasenv ((Some (nn, C.Decl s))::context) ugraph
            table (lift_amount+1) t in
        let l1' =
          List.map
            (fun (t, s, m, ug, eq_found) ->
               C.Prod (nn, t, lifted_t), s, m, ug, eq_found) l1
        and l2' =
          List.map
            (fun (t, s, m, ug, eq_found) ->
               C.Prod (nn, lifted_s, t), s, m, ug, eq_found) l2 in
        l1' @ l2', C.Prod (nn, lifted_s, lifted_t)
          
    | C.Lambda (nn, s, t) ->
        let l1, lifted_s =
          betaexpand_term metasenv context ugraph table lift_amount s in
        let l2, lifted_t =
          betaexpand_term metasenv ((Some (nn, C.Decl s))::context) ugraph
            table (lift_amount+1) t in
        let l1' =
          List.map
            (fun (t, s, m, ug, eq_found) ->
               C.Lambda (nn, t, lifted_t), s, m, ug, eq_found) l1
        and l2' =
          List.map
            (fun (t, s, m, ug, eq_found) ->
               C.Lambda (nn, lifted_s, t), s, m, ug, eq_found) l2 in
        l1' @ l2', C.Lambda (nn, lifted_s, lifted_t)

    | C.Appl l ->
        let l', lifted_l =
          List.fold_right
            (fun arg (res, lifted_tl) ->
               let arg_res, lifted_arg =
                 betaexpand_term metasenv context ugraph table lift_amount arg
               in
               let l1 =
                 List.map
                   (fun (a, s, m, ug, eq_found) ->
                      a::lifted_tl, s, m, ug, eq_found)
                   arg_res
               in
               (l1 @
                  (List.map
                     (fun (r, s, m, ug, eq_found) ->
                        lifted_arg::r, s, m, ug, eq_found)
                     res),
                lifted_arg::lifted_tl)
            ) l ([], [])
        in
        (List.map
           (fun (l, s, m, ug, eq_found) -> (C.Appl l, s, m, ug, eq_found)) l',
         C.Appl lifted_l)

    | t -> [], (S.lift lift_amount t)
  in
  match term with
  | C.Meta (i, l) -> res, lifted_term
  | term ->
      let termty, ugraph =
        C.Implicit None, ugraph
(*         CicTypeChecker.type_of_aux' metasenv context term ugraph *)
      in
      let r = 
        find_all_matches
          metasenv context ugraph lift_amount term termty candidates
      in
      r @ res, lifted_term
;;


let sup_l_counter = ref 1;;

let superposition_left newmeta (metasenv, context, ugraph) table target =
  let module C = Cic in
  let module S = CicSubstitution in
  let module M = CicMetaSubst in
  let module HL = HelmLibraryObjects in
  let module CR = CicReduction in
  let module U = Utils in
  let weight, proof, (eq_ty, left, right, ordering), _, _ = target in
  let expansions, _ =
    let term = if ordering = U.Gt then left else right in
    betaexpand_term metasenv context ugraph table 0 term
  in
  let maxmeta = ref newmeta in
  let build_new (bo, s, m, ug, (eq_found, eq_URI)) =

    print_endline "\nSUPERPOSITION LEFT\n";
    
    let time1 = Unix.gettimeofday () in
    
    let pos, (_, proof', (ty, what, other, _), menv', args') = eq_found in
    let what, other = if pos = Utils.Left then what, other else other, what in
    let newgoal, newproof =
      let bo' = (* M. *)apply_subst s (S.subst other bo) in
(*       let t' = *)
(*         let name = C.Name ("x_SupL_" ^ (string_of_int !sup_l_counter)) in *)
(*         incr sup_l_counter; *)
(*         let l, r = *)
(*           if ordering = U.Gt then bo, S.lift 1 right else S.lift 1 left, bo in *)
(*         (name, ty, S.lift 1 eq_ty, l, r) *)
(*       in *)
      let name = C.Name ("x_SupL_" ^ (string_of_int !sup_l_counter)) in
      incr sup_l_counter;
      let bo'' = 
        let l, r =
          if ordering = U.Gt then bo, S.lift 1 right else S.lift 1 left, bo in
        C.Appl [C.MutInd (LibraryObjects.eq_URI (), 0, []);
                S.lift 1 eq_ty; l; r]
      in
      incr maxmeta;
      let metaproof =
        let irl =
          CicMkImplicit.identity_relocation_list_for_metavariable context in
        C.Meta (!maxmeta, irl)
      in
(*       let target' = *)
        let eq_found =
          let proof' =
(*             let ens = *)
(*               if pos = Utils.Left then *)
(*                 build_ens_for_sym_eq ty what other *)
(*               else *)
(*                 build_ens_for_sym_eq ty other what *)
(*             in *)
            let termlist =
              if pos = Utils.Left then [ty; what; other]
              else [ty; other; what]
            in
            Inference.ProofSymBlock (termlist, proof')
          in
          let what, other =
            if pos = Utils.Left then what, other else other, what
          in
          pos, (0, proof', (ty, other, what, Utils.Incomparable), menv', args')
        in
        let target_proof =
          let pb =
            Inference.ProofBlock (s, eq_URI, (name, ty), bo''(* t' *), eq_found,
                                  Inference.BasicProof metaproof)
          in
          match proof with
          | Inference.BasicProof _ ->
              print_endline "replacing a BasicProof";
              pb
          | Inference.ProofGoalBlock (_, parent_proof(* parent_eq *)) ->
              print_endline "replacing another ProofGoalBlock";
              Inference.ProofGoalBlock (pb, parent_proof(* parent_eq *))
          | _ -> assert false
        in
(*         (weight, target_proof, (eq_ty, left, right, ordering), [], []) *)
(*       in *)
      let refl =
        C.Appl [C.MutConstruct (* reflexivity *)
                  (LibraryObjects.eq_URI (), 0, 1, []);
                eq_ty; if ordering = U.Gt then right else left]
      in
      (bo',
       Inference.ProofGoalBlock (Inference.BasicProof refl, target_proof(* target' *)))
    in
    let left, right =
      if ordering = U.Gt then newgoal, right else left, newgoal in
    let neworder = !Utils.compare_terms left right in

    let time2 = Unix.gettimeofday () in
    build_newtarget_time := !build_newtarget_time +. (time2 -. time1);

    let res =
      let w = Utils.compute_equality_weight eq_ty left right in
      (w, newproof, (eq_ty, left, right, neworder), [], [])
    in
    res
  in
  !maxmeta, List.map build_new expansions
;;


let sup_r_counter = ref 1;;

let superposition_right newmeta (metasenv, context, ugraph) table target =
  let module C = Cic in
  let module S = CicSubstitution in
  let module M = CicMetaSubst in
  let module HL = HelmLibraryObjects in
  let module CR = CicReduction in
  let module U = Utils in
  let _, eqproof, (eq_ty, left, right, ordering), newmetas, args = target in
  let metasenv' = metasenv @ newmetas in
  let maxmeta = ref newmeta in
  let res1, res2 =
    match ordering with
    | U.Gt -> fst (betaexpand_term metasenv' context ugraph table 0 left), []
    | U.Lt -> [], fst (betaexpand_term metasenv' context ugraph table 0 right)
    | _ ->
        let res l r =
          List.filter
            (fun (_, subst, _, _, _) ->
               let subst = (* M. *)apply_subst subst in
               let o = !Utils.compare_terms (subst l) (subst r) in
               o <> U.Lt && o <> U.Le)
            (fst (betaexpand_term metasenv' context ugraph table 0 l))
        in
        (res left right), (res right left)
  in
  let build_new ordering (bo, s, m, ug, (eq_found, eq_URI)) =

    let time1 = Unix.gettimeofday () in
    
    let pos, (_, proof', (ty, what, other, _), menv', args') = eq_found in
    let what, other = if pos = Utils.Left then what, other else other, what in
    let newgoal, newproof =
      let bo' = (* M. *)apply_subst s (S.subst other bo) in
      let t' =
        let name = C.Name ("x_SupR_" ^ (string_of_int !sup_r_counter)) in
        incr sup_r_counter;
        let l, r =
          if ordering = U.Gt then bo, S.lift 1 right else S.lift 1 left, bo in
        (name, ty, S.lift 1 eq_ty, l, r)
      in
      let name = C.Name ("x_SupR_" ^ (string_of_int !sup_r_counter)) in
      incr sup_r_counter;
      let bo'' =
        let l, r =
          if ordering = U.Gt then bo, S.lift 1 right else S.lift 1 left, bo in
        C.Appl [C.MutInd (LibraryObjects.eq_URI (), 0, []);
                S.lift 1 eq_ty; l; r]
      in
      bo',
      Inference.ProofBlock (
        s, eq_URI, (name, ty), bo''(* t' *), eq_found, eqproof)
    in
    let newmeta, newequality = 
      let left, right =
        if ordering = U.Gt then newgoal, (* M. *)apply_subst s right
        else (* M. *)apply_subst s left, newgoal in
      let neworder = !Utils.compare_terms left right 
      and newmenv = newmetas @ menv'
      and newargs = args @ args' in
(*         let m = *)
(*           (Inference.metas_of_term left) @ (Inference.metas_of_term right) in *)
(*         let a =  *)
(*           List.filter *)
(*             (function C.Meta (i, _) -> List.mem i m | _ -> assert false) *)
(*             (args @ args') *)
(*         in *)
(*         let delta = (List.length args) - (List.length a) in *)
(*         if delta > 0 then *)
(*           let first = List.hd a in *)
(*           let rec aux l = function *)
(*             | 0 -> l *)
(*             | d -> let l = aux l (d-1) in l @ [first] *)
(*           in *)
(*           aux a delta *)
(*         else *)
(*           a *)
(*       in *)
      let eq' =
        let w = Utils.compute_equality_weight eq_ty left right in
        (w, newproof, (eq_ty, left, right, neworder), newmenv, newargs)
      and env = (metasenv, context, ugraph) in
      let newm, eq' = Inference.fix_metas !maxmeta eq' in
      newm, eq'
    in
    maxmeta := newmeta;

    let time2 = Unix.gettimeofday () in
    build_newtarget_time := !build_newtarget_time +. (time2 -. time1);

    newequality
  in
  let new1 = List.map (build_new U.Gt) res1
  and new2 = List.map (build_new U.Lt) res2 in
(*   let ok = function *)
(*     | _, _, (_, left, right, _), _, _ -> *)
(*         not (fst (CR.are_convertible context left right ugraph)) *)
(*   in *)
  let _ =
    let env = metasenv, context, ugraph in
    debug_print
      (lazy
         (Printf.sprintf "end of superposition_right:\n%s\n"
            (String.concat "\n"
               ((List.map
                   (fun e -> "Positive " ^
                      (Inference.string_of_equality ~env e)) (new1 @ new2))))))
  in
  let ok e = not (Inference.is_identity (metasenv, context, ugraph) e) in
  (!maxmeta,
   (List.filter ok (new1 @ new2)))
;;


let rec demodulation_goal newmeta env table goal =
  let module C = Cic in
  let module S = CicSubstitution in
  let module M = CicMetaSubst in
  let module HL = HelmLibraryObjects in
  let metasenv, context, ugraph = env in
  let maxmeta = ref newmeta in
  let proof, metas, term = goal in
  let metasenv' = metasenv @ metas in

  let build_newgoal (t, subst, menv, ug, (eq_found, eq_URI)) =
    let pos, (_, proof', (ty, what, other, _), menv', args') = eq_found in
    let what, other = if pos = Utils.Left then what, other else other, what in
    let ty =
      try fst (CicTypeChecker.type_of_aux' metasenv context what ugraph)
      with CicUtil.Meta_not_found _ -> ty
    in
    let newterm, newproof =
      let bo = (* M. *)apply_subst subst (S.subst other t) in
      let bo' = apply_subst subst t in 
      let name = C.Name ("x_DemodGoal_" ^ (string_of_int !demod_counter)) in
      incr demod_counter;
      let metaproof = 
        incr maxmeta;
        let irl =
          CicMkImplicit.identity_relocation_list_for_metavariable context in
        Printf.printf "\nADDING META: %d\n" !maxmeta;
        print_newline ();
        C.Meta (!maxmeta, irl)
      in
      let eq_found =
        let proof' =
(*           let ens = *)
(*             if pos = Utils.Left then build_ens_for_sym_eq ty what other *)
(*             else build_ens_for_sym_eq ty other what *)
(*           in *)
          let termlist =
            if pos = Utils.Left then [ty; what; other]
            else [ty; other; what]
          in
          Inference.ProofSymBlock (termlist, proof')
        in
        let what, other =
          if pos = Utils.Left then what, other else other, what
        in
        pos, (0, proof', (ty, other, what, Utils.Incomparable), menv', args')
      in
      let goal_proof =
        let pb =
          Inference.ProofBlock (subst, eq_URI, (name, ty), bo',
                                eq_found, Inference.BasicProof metaproof)
        in
        let rec repl = function
          | Inference.NoProof ->
              debug_print (lazy "replacing a NoProof");
              pb
          | Inference.BasicProof _ ->
              debug_print (lazy "replacing a BasicProof");
              pb
          | Inference.ProofGoalBlock (_, parent_proof) ->
              debug_print (lazy "replacing another ProofGoalBlock");
              Inference.ProofGoalBlock (pb, parent_proof)
          | (Inference.SubProof (term, meta_index, p) as subproof) ->
              debug_print
                (lazy
                   (Printf.sprintf "replacing %s"
                      (Inference.string_of_proof subproof)));
              Inference.SubProof (term, meta_index, repl p)
          | _ -> assert false
        in repl proof
      in
      bo, Inference.ProofGoalBlock (Inference.NoProof, goal_proof)
    in
    let m = Inference.metas_of_term newterm in
    let newmetasenv = List.filter (fun (i, _, _) -> List.mem i m) metas in
    !maxmeta, (newproof, newmetasenv, newterm)
  in  
  let res = demodulation_aux metasenv' context ugraph table 0 term in
  match res with
  | Some t ->
      let newmeta, newgoal = build_newgoal t in
      let _, _, newg = newgoal in
      if Inference.meta_convertibility term newg then
        newmeta, newgoal
      else
        demodulation_goal newmeta env table newgoal
  | None ->
      newmeta, goal
;;


let rec demodulation_theorem newmeta env table theorem =
  let module C = Cic in
  let module S = CicSubstitution in
  let module M = CicMetaSubst in
  let module HL = HelmLibraryObjects in
  let metasenv, context, ugraph = env in
  let maxmeta = ref newmeta in
  let proof, metas, term = theorem in
  let term, termty, metas = theorem in
  let metasenv' = metasenv @ metas in

  let build_newtheorem (t, subst, menv, ug, (eq_found, eq_URI)) =
    let pos, (_, proof', (ty, what, other, _), menv', args') = eq_found in
    let what, other = if pos = Utils.Left then what, other else other, what in
    let newterm, newty =
      let bo = apply_subst subst (S.subst other t) in
      let bo' = apply_subst subst t in 
      let name = C.Name ("x_DemodThm_" ^ (string_of_int !demod_counter)) in
      incr demod_counter;
      let newproof =
        Inference.ProofBlock (subst, eq_URI, (name, ty), bo', eq_found,
                              Inference.BasicProof term)
      in
      (Inference.build_proof_term newproof, bo)
    in
    let m = Inference.metas_of_term newterm in
    let newmetasenv = List.filter (fun (i, _, _) -> List.mem i m) metas in
    !maxmeta, (newterm, newty, newmetasenv)
  in  
  let res = demodulation_aux metasenv' context ugraph table 0 termty in
  match res with
  | Some t ->
      let newmeta, newthm = build_newtheorem t in
      let newt, newty, _ = newthm in
      if Inference.meta_convertibility termty newty then
        newmeta, newthm
      else
        demodulation_theorem newmeta env table newthm
  | None ->
      newmeta, theorem
;;