(* 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/.
 *)

(* $Id: inference.ml 6245 2006-04-05 12:07:51Z tassi $ *)


(******* CIC substitution ***************************************************)

type cic_substitution = Cic.substitution
let cic_apply_subst = CicMetaSubst.apply_subst
let cic_apply_subst_metasenv = CicMetaSubst.apply_subst_metasenv
let cic_ppsubst = CicMetaSubst.ppsubst
let cic_buildsubst n context t ty tail = (n,(context,t,ty)) :: tail
let cic_flatten_subst subst =
    List.map
      (fun (i, (context, term, ty)) ->
         let context = (* cic_apply_subst_context subst*) context in
         let term = cic_apply_subst subst term in
         let ty = cic_apply_subst subst ty in  
         (i, (context, term, ty))) subst
let rec cic_lookup_subst meta subst =
  match meta with
  | Cic.Meta (i, _) -> (
      try let _, (_, t, _) = List.find (fun (m, _) -> m = i) subst 
      in cic_lookup_subst t subst 
      with Not_found -> meta
    )
  | _ -> meta
;;

let cic_merge_subst_if_possible s1 s2 =
  let already_in = Hashtbl.create 13 in
  let rec aux acc = function
    | ((i,_,x) as s)::tl ->
        (try 
          let x' = Hashtbl.find already_in i in
          if x = x' then aux acc tl else None
        with
        | Not_found -> 
            Hashtbl.add already_in i x;
            aux (s::acc) tl)
    | [] -> Some acc 
  in  
    aux [] (s1@s2)
;;

(******** NAIF substitution **************************************************)
(* 
 * naif version of apply subst; the local context of metas is ignored;
 * we assume the substituted term must be lifted according to the nesting
 * depth of the meta. 
 * Alternatively, we could used implicit instead of metas 
 *)

type naif_substitution = (int * Cic.term) list 

let naif_apply_subst subst term =
 let rec aux k t =
   match t with
       Cic.Rel _ -> t
     | Cic.Var (uri,exp_named_subst) -> 
         let exp_named_subst' =
           List.map (fun (uri, t) -> (uri, aux k t)) exp_named_subst
         in
           Cic.Var (uri, exp_named_subst')
    | Cic.Meta (i, l) -> 
        (try
          aux k (CicSubstitution.lift k (List.assoc i subst)) 
         with Not_found -> t)
    | Cic.Sort _
    | Cic.Implicit _ -> t
    | Cic.Cast (te,ty) -> Cic.Cast (aux k te, aux k ty)
    | Cic.Prod (n,s,t) -> Cic.Prod (n, aux k s, aux (k+1) t)
    | Cic.Lambda (n,s,t) -> Cic.Lambda (n, aux k s, aux (k+1) t)
    | Cic.LetIn (n,s,t) -> Cic.LetIn (n, aux k s, aux (k+1) t)
    | Cic.Appl [] -> assert false
    | Cic.Appl l -> Cic.Appl (List.map (aux k) l)
    | Cic.Const (uri,exp_named_subst) ->
        let exp_named_subst' =
          List.map (fun (uri, t) -> (uri, aux k t)) exp_named_subst
        in
          if exp_named_subst' != exp_named_subst then
            Cic.Const (uri, exp_named_subst')
          else
            t (* TODO: provare a mantenere il piu' possibile sharing *)
    | Cic.MutInd (uri,typeno,exp_named_subst) ->
        let exp_named_subst' =
          List.map (fun (uri, t) -> (uri, aux k t)) exp_named_subst
        in
          Cic.MutInd (uri,typeno,exp_named_subst')
    | Cic.MutConstruct (uri,typeno,consno,exp_named_subst) ->
        let exp_named_subst' =
          List.map (fun (uri, t) -> (uri, aux k t)) exp_named_subst
        in
          Cic.MutConstruct (uri,typeno,consno,exp_named_subst')
    | Cic.MutCase (sp,i,outty,t,pl) ->
        let pl' = List.map (aux k) pl in
          Cic.MutCase (sp, i, aux k outty, aux k t, pl')
    | Cic.Fix (i, fl) ->
        let len = List.length fl in
        let fl' =
         List.map 
           (fun (name, i, ty, bo) -> (name, i, aux k ty, aux (k+len) bo)) fl
        in
          Cic.Fix (i, fl')
    | Cic.CoFix (i, fl) ->
        let len = List.length fl in
        let fl' =
          List.map (fun (name, ty, bo) -> (name, aux k ty, aux (k+len) bo)) fl
        in
          Cic.CoFix (i, fl')
in
  aux 0 term
;;

(* naif version of apply_subst_metasenv: we do not apply the 
substitution to the context *)

let naif_apply_subst_metasenv subst metasenv =
  List.map
    (fun (n, context, ty) ->
      (n, context, naif_apply_subst subst ty))
    (List.filter
      (fun (i, _, _) -> not (List.mem_assoc i subst))
      metasenv)

let naif_ppsubst names subst =
  "{" ^ String.concat "; "
    (List.map
      (fun (idx, t) ->
         Printf.sprintf "%d:= %s" idx (CicPp.pp t names))
    subst) ^ "}"
;;

let naif_buildsubst n context t ty tail = (n,t) :: tail ;;

let naif_flatten_subst subst = 
  List.map (fun (i,t) -> i, naif_apply_subst subst t ) subst
;;

let rec naif_lookup_subst meta subst =
  match meta with
    | Cic.Meta (i, _) ->
        (try
          naif_lookup_subst (List.assoc i subst) subst
        with
            Not_found -> meta)
    | _ -> meta
;;

let naif_merge_subst_if_possible s1 s2 =
  let already_in = Hashtbl.create 13 in
  let rec aux acc = function
    | ((i,x) as s)::tl ->
        (try 
          let x' = Hashtbl.find already_in i in
          if x = x' then aux acc tl else None
        with
        | Not_found -> 
            Hashtbl.add already_in i x;
            aux (s::acc) tl)
    | [] -> Some acc 
  in  
    aux [] (s1@s2)
;;

(********** ACTUAL SUBSTITUTION IMPLEMENTATION *******************************)

type substitution = naif_substitution
let apply_subst = naif_apply_subst
let apply_subst_metasenv = naif_apply_subst_metasenv
let ppsubst ~names l = naif_ppsubst (names:(Cic.name option)list) l
let buildsubst = naif_buildsubst
let flatten_subst = naif_flatten_subst
let lookup_subst = naif_lookup_subst

(* filter out from metasenv the variables in substs *)
let filter subst metasenv =
  List.filter
    (fun (m, _, _) ->
         try let _ = List.find (fun (i, _) -> m = i) subst in false
         with Not_found -> true)
    metasenv
;;

let is_in_subst i subst = List.mem_assoc i subst;;
  
let merge_subst_if_possible = naif_merge_subst_if_possible;;

let empty_subst = [];;

(********* EQUALITY **********************************************************)

type rule = SuperpositionRight | SuperpositionLeft | Demodulation
type uncomparable = int -> int 
type equality =
    uncomparable *       (* trick to break structural equality *)
    int  *               (* weight *)
    proof * 
    (Cic.term *          (* type *)
     Cic.term *          (* left side *)
     Cic.term *          (* right side *)
     Utils.comparison) * (* ordering *)  
    Cic.metasenv  *      (* environment for metas *)
    int                  (* id *)
and proof = new_proof * old_proof 

and new_proof = 
  | Exact of Cic.term
  | Step of substitution * (rule * int*(Utils.pos*int)* Cic.term) (* eq1, eq2,predicate *)  
and old_proof =
  | NoProof (* term is the goal missing a proof *)
  | BasicProof of substitution * Cic.term
  | ProofBlock of
      substitution * UriManager.uri *
        (Cic.name * Cic.term) * Cic.term * (Utils.pos * equality) * old_proof
  | ProofGoalBlock of old_proof * old_proof 
  | ProofSymBlock of Cic.term list * old_proof
  | SubProof of Cic.term * int * old_proof
and goal_proof = (Utils.pos * int * substitution * Cic.term) list
;;

(* globals *)
let maxid = ref 0;;
let id_to_eq = Hashtbl.create 1024;;

let freshid () =
  incr maxid; !maxid
;;

let reset () = 
  maxid := 0;
  Hashtbl.clear  id_to_eq
;;

let uncomparable = fun _ -> 0

let mk_equality (weight,(newp,oldp),(ty,l,r,o),m) =
  let id = freshid () in
  let eq = (uncomparable,weight,(newp,oldp),(ty,l,r,o),m,id) in
  Hashtbl.add id_to_eq id eq;
  eq
;;

let open_equality (_,weight,proof,(ty,l,r,o),m,id) = 
  (weight,proof,(ty,l,r,o),m,id)

let string_of_equality ?env eq =
  match env with
  | None ->
      let w, _, (ty, left, right, o), _ , id = open_equality eq in
      Printf.sprintf "Id: %d, Weight: %d, {%s}: %s =(%s) %s" 
              id w (CicPp.ppterm ty)
              (CicPp.ppterm left) 
              (Utils.string_of_comparison o) (CicPp.ppterm right)
  | Some (_, context, _) -> 
      let names = Utils.names_of_context context in
      let w, _, (ty, left, right, o), _ , id = open_equality eq in
      Printf.sprintf "Id: %d, Weight: %d, {%s}: %s =(%s) %s" 
              id w (CicPp.pp ty names)
              (CicPp.pp left names) (Utils.string_of_comparison o)
              (CicPp.pp right names)
;;

let compare (_,_,_,s1,_,_) (_,_,_,s2,_,_) =
  Pervasives.compare s1 s2
;;

let rec string_of_proof_old ?(names=[]) = function
  | NoProof -> "NoProof " 
  | BasicProof (s, t) -> "BasicProof(" ^ 
      ppsubst ~names s ^ ", " ^ (CicPp.pp t names) ^ ")"
  | SubProof (t, i, p) ->
      Printf.sprintf "SubProof(%s, %s, %s)"
        (CicPp.pp t names) (string_of_int i) (string_of_proof_old p)
  | ProofSymBlock (_,p) -> 
      Printf.sprintf "ProofSymBlock(%s)" (string_of_proof_old p)
  | ProofBlock (subst, _, _, _ ,(_,eq),old) -> 
      let _,(_,p),_,_,_ = open_equality eq in 
      "ProofBlock(" ^ (ppsubst ~names subst) ^ "," ^ (string_of_proof_old old) ^ "," ^
      string_of_proof_old p ^ ")"
  | ProofGoalBlock (p1, p2) ->
      Printf.sprintf "ProofGoalBlock(%s, %s)"
        (string_of_proof_old p1) (string_of_proof_old p2)
;;


let proof_of_id id =
  try
    let (_,(p,_),(_,l,r,_),_,_) = open_equality (Hashtbl.find id_to_eq id) in
      p,l,r
  with
      Not_found -> assert false


let string_of_proof_new ?(names=[]) p gp = 
  let str_of_rule = function
    | SuperpositionRight -> "SupR"
    | SuperpositionLeft -> "SupL"
    | Demodulation -> "Demod"
  in
  let str_of_pos = function
    | Utils.Left -> "left"
    | Utils.Right -> "right"
  in
  let fst3 (x,_,_) = x in
  let rec aux margin name = 
    let prefix = String.make margin ' ' ^ name ^ ": " in function 
    | Exact t -> 
        Printf.sprintf "%sExact (%s)\n" 
          prefix (CicPp.pp t names)
    | Step (subst,(rule,eq1,(pos,eq2),pred)) -> 
        Printf.sprintf "%s%s(%s|%d with %d dir %s pred %s))\n"
          prefix (str_of_rule rule) (ppsubst ~names subst) eq1 eq2 (str_of_pos pos) 
          (CicPp.pp pred names)^ 
        aux (margin+1) (Printf.sprintf "%d" eq1) (fst3 (proof_of_id eq1)) ^ 
        aux (margin+1) (Printf.sprintf "%d" eq2) (fst3 (proof_of_id eq2)) 
  in
  aux 0 "" p ^ 
  String.concat "\n" 
    (List.map 
      (fun (pos,i,s,t) -> 
        (Printf.sprintf 
          "GOAL: %s %d %s %s\n" 
            (str_of_pos pos) i (ppsubst ~names s) (CicPp.pp t names)) ^ 
        aux 1 (Printf.sprintf "%d " i) (fst3 (proof_of_id i)))
      gp)
;;

let ppsubst = ppsubst ~names:[]

(* returns an explicit named subst and a list of arguments for sym_eq_URI *)
let build_ens uri termlist =
  let obj, _ = CicEnvironment.get_obj CicUniv.empty_ugraph uri in
  match obj with
  | Cic.Constant (_, _, _, uris, _) ->
      assert (List.length uris <= List.length termlist);
      let rec aux = function
        | [], tl -> [], tl
        | (uri::uris), (term::tl) ->
            let ens, args = aux (uris, tl) in
            (uri, term)::ens, args
        | _, _ -> assert false
      in
      aux (uris, termlist)
  | _ -> assert false
;;

let build_proof_term_old ?(noproof=Cic.Implicit None) proof =
  let rec do_build_proof proof = 
    match proof with
    | NoProof ->
        Printf.fprintf stderr "WARNING: no proof!\n";
        noproof
    | BasicProof (s,term) -> apply_subst s term
    | ProofGoalBlock (proofbit, proof) ->
        print_endline "found ProofGoalBlock, going up...";
        do_build_goal_proof proofbit proof
    | ProofSymBlock (termlist, proof) ->
        let proof = do_build_proof proof in
        let ens, args = build_ens (Utils.sym_eq_URI ()) termlist in
        Cic.Appl ([Cic.Const (Utils.sym_eq_URI (), ens)] @ args @ [proof])
    | ProofBlock (subst, eq_URI, (name, ty), bo, (pos, eq), eqproof) ->
        let t' = Cic.Lambda (name, ty, bo) in
        let _, (_,proof), (ty, what, other, _), menv',_ = open_equality eq in
        let proof' = do_build_proof proof in
        let eqproof = do_build_proof eqproof in
        let what, other =
          if pos = Utils.Left then what, other else other, what
        in
        apply_subst subst
          (Cic.Appl [Cic.Const (eq_URI, []); ty;
                     what; t'; eqproof; other; proof'])
    | SubProof (term, meta_index, proof) ->
        let proof = do_build_proof proof in
        let eq i = function
          | Cic.Meta (j, _) -> i = j
          | _ -> false
        in
        ProofEngineReduction.replace
          ~equality:eq ~what:[meta_index] ~with_what:[proof] ~where:term

  and do_build_goal_proof proofbit proof =
    match proof with
    | ProofGoalBlock (pb, p) ->
        do_build_proof (ProofGoalBlock (replace_proof proofbit pb, p))
    | _ -> do_build_proof (replace_proof proofbit proof)

  and replace_proof newproof = function
    | ProofBlock (subst, eq_URI, namety, bo, poseq, eqproof) ->
        let eqproof' = replace_proof newproof eqproof in
        ProofBlock (subst, eq_URI, namety, bo, poseq, eqproof')
    | ProofGoalBlock (pb, p) ->
        let pb' = replace_proof newproof pb in
        ProofGoalBlock (pb', p)
    | BasicProof _ -> newproof
    | SubProof (term, meta_index, p) ->
        SubProof (term, meta_index, replace_proof newproof p)
    | p -> p
  in
  do_build_proof proof 
;;

let mk_sym uri ty t1 t2 p =
  let ens, args =  build_ens uri [ty;t1;t2;p] in
    Cic.Appl (Cic.Const(uri, ens) :: args)
;;

let mk_trans uri ty t1 t2 t3 p12 p23 =
  let ens, args = build_ens uri [ty;t1;t2;t3;p12;p23] in
    Cic.Appl (Cic.Const (uri, ens) :: args)
;;

let mk_eq_ind uri ty what pred p1 other p2 =
 Cic.Appl [Cic.Const (uri, []); ty; what; pred; p1; other; p2]
;;

let p_of_sym ens tl =
  let args = List.map snd ens @ tl in
  match args with 
    | [_;_;_;p] -> p 
    | _ -> assert false 
;;

let open_trans ens tl =
  let args = List.map snd ens @ tl in
  match args with 
    | [ty;l;m;r;p1;p2] -> ty,l,m,r,p1,p2
    | _ -> assert false   
;;

let canonical t = 
  let rec remove_refl t =
    match t with
    | Cic.Appl (((Cic.Const(uri_trans,ens))::tl) as args)
          when LibraryObjects.is_trans_eq_URI uri_trans ->
          let ty,l,m,r,p1,p2 = open_trans ens tl in
            (match p1,p2 with
              | Cic.Appl [Cic.MutConstruct (uri, 0, 1,_);_;_],p2 -> 
                  remove_refl p2
              | p1,Cic.Appl [Cic.MutConstruct (uri, 0, 1,_);_;_] -> 
                  remove_refl p1
              | _ -> Cic.Appl (List.map remove_refl args))
    | Cic.Appl l -> Cic.Appl (List.map remove_refl l)
    | _ -> t
  in
  let rec canonical t =
    match t with
      | Cic.Appl (((Cic.Const(uri_sym,ens))::tl) as args)
          when LibraryObjects.is_sym_eq_URI uri_sym ->
          (match p_of_sym ens tl with
             | Cic.Appl ((Cic.Const(uri,ens))::tl)
                 when LibraryObjects.is_sym_eq_URI uri -> 
                   canonical (p_of_sym ens tl)
             | Cic.Appl ((Cic.Const(uri_trans,ens))::tl)
                 when LibraryObjects.is_trans_eq_URI uri_trans ->
                 let ty,l,m,r,p1,p2 = open_trans ens tl in
                   mk_trans uri_trans ty r m l 
                     (canonical (mk_sym uri_sym ty m r p2)) 
                     (canonical (mk_sym uri_sym ty l m p1))
             | Cic.Appl (((Cic.Const(uri_ind,ens)) as he)::tl) 
                 when LibraryObjects.is_eq_ind_URI uri_ind || 
                      LibraryObjects.is_eq_ind_r_URI uri_ind ->
                 let ty, what, pred, p1, other, p2 =
                   match tl with
                   | [ty;what;pred;p1;other;p2] -> ty, what, pred, p1, other, p2
                   | _ -> assert false
                 in
                 let pred,l,r = 
                   match pred with
                   | Cic.Lambda (name,s,Cic.Appl [Cic.MutInd(uri,0,ens);ty;l;r])
                       when LibraryObjects.is_eq_URI uri ->
                         Cic.Lambda 
                           (name,s,Cic.Appl [Cic.MutInd(uri,0,ens);ty;r;l]),l,r
                   | _ -> 
                       prerr_endline (CicPp.ppterm pred);
                       assert false
                 in
                 let l = CicSubstitution.subst what l in
                 let r = CicSubstitution.subst what r in
                 Cic.Appl 
                   [he;ty;what;pred;
                    canonical (mk_sym uri_sym ty l r p1);other;canonical p2]
             | Cic.Appl [Cic.MutConstruct (uri, 0, 1,_);_;_] as t
                 when LibraryObjects.is_eq_URI uri -> t
             | _ -> Cic.Appl (List.map canonical args))
      | Cic.Appl l -> Cic.Appl (List.map canonical l)
      | _ -> t
  in
  remove_refl (canonical t)  
;;

let build_proof_step subst p1 p2 pos l r pred =
  let p1 = apply_subst subst p1 in
  let p2 = apply_subst subst p2 in
  let l = apply_subst subst l in
  let r = apply_subst subst r in
  let pred = apply_subst subst pred in
  let ty,body = (* Cic.Implicit None *) 
    match pred with
      | Cic.Lambda (_,ty,body) -> ty,body 
      | _ -> assert false
  in
  let what, other = (* Cic.Implicit None, Cic.Implicit None *)
    if pos = Utils.Left then l,r else r,l
  in
  let is_not_fixed t =
   CicSubstitution.subst (Cic.Implicit None) t <>
   CicSubstitution.subst (Cic.Rel 1) t
  in
    match body,pos with
      |Cic.Appl [Cic.MutInd(eq,_,_);_;Cic.Rel 1;third],Utils.Left ->
         let third = CicSubstitution.subst (Cic.Implicit None) third in
         let uri_trans = LibraryObjects.trans_eq_URI ~eq in
         let uri_sym = LibraryObjects.sym_eq_URI ~eq in
           mk_trans uri_trans ty other what third
            (mk_sym uri_sym ty what other p2) p1
      |Cic.Appl [Cic.MutInd(eq,_,_);_;Cic.Rel 1;third],Utils.Right ->
         let third = CicSubstitution.subst (Cic.Implicit None) third in
         let uri_trans = LibraryObjects.trans_eq_URI ~eq in
           mk_trans uri_trans ty other what third p2 p1
      |Cic.Appl [Cic.MutInd(eq,_,_);_;third;Cic.Rel 1],Utils.Left -> 
         let third = CicSubstitution.subst (Cic.Implicit None) third in
         let uri_trans = LibraryObjects.trans_eq_URI ~eq in
           mk_trans uri_trans ty third what other p1 p2  
      |Cic.Appl [Cic.MutInd(eq,_,_);_;third;Cic.Rel 1],Utils.Right -> 
         let third = CicSubstitution.subst (Cic.Implicit None) third in
         let uri_trans = LibraryObjects.trans_eq_URI ~eq in
         let uri_sym = LibraryObjects.sym_eq_URI ~eq in
           mk_trans uri_trans ty third what other p1
            (mk_sym uri_sym ty other what p2)
      | Cic.Appl [Cic.MutInd(eq,_,_);_;lhs;rhs],Utils.Left when is_not_fixed lhs
        ->
          let rhs = CicSubstitution.subst (Cic.Implicit None) rhs in
          let uri_trans = LibraryObjects.trans_eq_URI ~eq in
          let uri_sym = LibraryObjects.sym_eq_URI ~eq in
          let pred_of t = CicSubstitution.subst t lhs in
          let pred_of_what = pred_of what in
          let pred_of_other = pred_of other in
          (*           p2 : what = other
           * ====================================
           *  inject p2:  P(what) = P(other)
           *)
          let inject ty lhs what other p2 =
           let liftedty = CicSubstitution.lift 1 ty in
           let lifted_pred_of_other = CicSubstitution.lift 1 (pred_of other) in
           let refl_eq_part =
            Cic.Appl [Cic.MutConstruct(eq,0,1,[]);ty;pred_of other]
           in
            mk_eq_ind (Utils.eq_ind_r_URI ()) ty other
             (Cic.Lambda (Cic.Name "foo",ty,
               (Cic.Appl
                 [Cic.MutInd(eq,0,[]);liftedty;lhs;lifted_pred_of_other])))
                refl_eq_part what p2
          in
           mk_trans uri_trans ty pred_of_other pred_of_what rhs
            (mk_sym uri_sym ty pred_of_what pred_of_other
             (inject ty lhs what other p2)
            ) p1
      | Cic.Appl[Cic.MutInd(eq,_,_);_;lhs;rhs],Utils.Right when is_not_fixed lhs
        ->
          let rhs = CicSubstitution.subst (Cic.Implicit None) rhs in
          let uri_trans = LibraryObjects.trans_eq_URI ~eq in
          let uri_sym = LibraryObjects.sym_eq_URI ~eq in
          let pred_of t = CicSubstitution.subst t lhs in
          let pred_of_what = pred_of what in
          let pred_of_other = pred_of other in
          (*           p2 : what = other
           * ====================================
           *  inject p2:  P(what) = P(other)
           *)
          let inject ty lhs what other p2 =
           let liftedty = CicSubstitution.lift 1 ty in
           let lifted_pred_of_other = CicSubstitution.lift 1 (pred_of other) in
           let refl_eq_part =
            Cic.Appl [Cic.MutConstruct(eq,0,1,[]);ty;pred_of other]
           in
            mk_eq_ind (Utils.eq_ind_URI ()) ty other
             (Cic.Lambda (Cic.Name "foo",ty,
               (Cic.Appl
                 [Cic.MutInd(eq,0,[]);liftedty;lhs;lifted_pred_of_other])))
                refl_eq_part what p2
          in
           mk_trans uri_trans ty pred_of_other pred_of_what rhs
            (mk_sym uri_sym ty pred_of_other pred_of_what
             (inject ty lhs what other p2)
            ) p1
      | Cic.Appl[Cic.MutInd(eq,_,_);_;lhs;rhs],Utils.Right when is_not_fixed rhs
        ->
          let lhs = CicSubstitution.subst (Cic.Implicit None) lhs in
          let uri_trans = LibraryObjects.trans_eq_URI ~eq in
          let uri_sym = LibraryObjects.sym_eq_URI ~eq in
          let pred_of t = CicSubstitution.subst t rhs in
          let pred_of_what = pred_of what in
          let pred_of_other = pred_of other in
          (*           p2 : what = other
           * ====================================
           *  inject p2:  P(what) = P(other)
           *)
          let inject ty lhs what other p2 =
           let liftedty = CicSubstitution.lift 1 ty in
           let lifted_pred_of_other = CicSubstitution.lift 1 (pred_of other) in
           let refl_eq_part =
            Cic.Appl [Cic.MutConstruct(eq,0,1,[]);ty;pred_of other]
           in
            mk_eq_ind (Utils.eq_ind_r_URI ()) ty other
             (Cic.Lambda (Cic.Name "foo",ty,
               (Cic.Appl
                 [Cic.MutInd(eq,0,[]);liftedty;lhs;lifted_pred_of_other])))
                refl_eq_part what p2
          in
           mk_trans uri_trans ty lhs pred_of_what pred_of_other p1
            (mk_sym uri_sym ty pred_of_what pred_of_other
             (inject ty rhs other what p2))
      | Cic.Appl[Cic.MutInd(eq,_,_);_;lhs;rhs],Utils.Left when is_not_fixed rhs
        ->
          let lhs = CicSubstitution.subst (Cic.Implicit None) lhs in
          let uri_trans = LibraryObjects.trans_eq_URI ~eq in
          let uri_sym = LibraryObjects.sym_eq_URI ~eq in
          let pred_of t = CicSubstitution.subst t rhs in
          let pred_of_what = pred_of what in
          let pred_of_other = pred_of other in
          (*           p2 : what = other
           * ====================================
           *  inject p2:  P(what) = P(other)
           *)
          let inject ty lhs what other p2 =
           let liftedty = CicSubstitution.lift 1 ty in
           let lifted_pred_of_other = CicSubstitution.lift 1 (pred_of other) in
           let refl_eq_part =
            Cic.Appl [Cic.MutConstruct(eq,0,1,[]);ty;pred_of other]
           in
            mk_eq_ind (Utils.eq_ind_URI ()) ty other
             (Cic.Lambda (Cic.Name "foo",ty,
               (Cic.Appl
                 [Cic.MutInd(eq,0,[]);liftedty;lhs;lifted_pred_of_other])))
                refl_eq_part what p2
          in
           mk_trans uri_trans ty lhs pred_of_what pred_of_other p1
            (mk_sym uri_sym ty pred_of_other pred_of_what
             (inject ty rhs other what p2))
      | _, Utils.Left ->
        mk_eq_ind (Utils.eq_ind_URI ()) ty what pred p1 other p2
      | _, Utils.Right ->
        mk_eq_ind (Utils.eq_ind_r_URI ()) ty what pred p1 other p2
;;

let build_proof_term_new proof =
  let rec aux = function
     | Exact term -> term
     | Step (subst,(_, id1, (pos,id2), pred)) ->
         let p,_,_ = proof_of_id id1 in
         let p1 = aux p in
         let p,l,r = proof_of_id id2 in
         let p2 = aux p in
           build_proof_step subst p1 p2 pos l r pred
  in
   aux proof

let build_goal_proof l initial =
  let proof = 
   List.fold_left 
   (fun  current_proof (pos,id,subst,pred) -> 
      let p,l,r = proof_of_id id in
      let p = build_proof_term_new p in
      let pos = if pos = Utils.Left then Utils.Right else Utils.Left in
        build_proof_step subst current_proof p pos l r pred)
   initial l
  in
  canonical proof
;;

let refl_proof ty term = 
  Cic.Appl 
    [Cic.MutConstruct 
       (LibraryObjects.eq_URI (), 0, 1, []);
       ty; term]
;;

let metas_of_proof p = Utils.metas_of_term (build_proof_term_old (snd p)) ;;

let relocate newmeta menv =
  let subst, metasenv, newmeta = 
    List.fold_right 
      (fun (i, context, ty) (subst, menv, maxmeta) ->         
        let irl = [] (*
         CicMkImplicit.identity_relocation_list_for_metavariable context *)
        in
        let newsubst = buildsubst i context (Cic.Meta(maxmeta,irl)) ty subst in
        let newmeta = maxmeta, context, ty in
        newsubst, newmeta::menv, maxmeta+1) 
      menv ([], [], newmeta+1)
  in
  let metasenv = apply_subst_metasenv subst metasenv in
  let subst = flatten_subst subst in
  subst, metasenv, newmeta


let fix_metas newmeta eq = 
  let w, (p1,p2), (ty, left, right, o), menv,_ = open_equality eq in
  (* debug 
  let _ , eq = 
    fix_metas_old newmeta (w, p, (ty, left, right, o), menv, args) in
  prerr_endline (string_of_equality eq); *)
  let subst, metasenv, newmeta = relocate newmeta menv in
  let ty = apply_subst subst ty in
  let left = apply_subst subst left in
  let right = apply_subst subst right in
  let fix_proof = function
    | NoProof -> NoProof 
    | BasicProof (subst',term) -> BasicProof (subst@subst',term)
    | ProofBlock (subst', eq_URI, namety, bo, (pos, eq), p) ->
        (*
        let newsubst = 
          List.map
            (fun (i, (context, term, ty)) ->
               let context = apply_subst_context subst context in
               let term = apply_subst subst term in
               let ty = apply_subst subst ty in  
                 (i, (context, term, ty))) subst' in *)
          ProofBlock (subst@subst', eq_URI, namety, bo, (pos, eq), p)
    | p -> assert false
  in
  let fix_new_proof = function
    | Exact p -> Exact (apply_subst subst p)
    | Step (s,(r,id1,(pos,id2),pred)) -> 
        Step (s@subst,(r,id1,(pos,id2),(*apply_subst subst*) pred))
  in
  let new_p = fix_new_proof p1 in
  let old_p = fix_proof p2 in
  let eq = mk_equality (w, (new_p,old_p), (ty, left, right, o), metasenv) in
  (* debug prerr_endline (string_of_equality eq); *)
  newmeta+1, eq  

exception NotMetaConvertible;;

let meta_convertibility_aux table t1 t2 =
  let module C = Cic in
  let rec aux ((table_l, table_r) as table) t1 t2 =
    match t1, t2 with
    | C.Meta (m1, tl1), C.Meta (m2, tl2) ->
        let m1_binding, table_l =
          try List.assoc m1 table_l, table_l
          with Not_found -> m2, (m1, m2)::table_l
        and m2_binding, table_r =
          try List.assoc m2 table_r, table_r
          with Not_found -> m1, (m2, m1)::table_r
        in
        if (m1_binding <> m2) || (m2_binding <> m1) then
          raise NotMetaConvertible
        else (
          try
            List.fold_left2
              (fun res t1 t2 ->
                 match t1, t2 with
                 | None, Some _ | Some _, None -> raise NotMetaConvertible
                 | None, None -> res
                 | Some t1, Some t2 -> (aux res t1 t2))
              (table_l, table_r) tl1 tl2
          with Invalid_argument _ ->
            raise NotMetaConvertible
        )
    | C.Var (u1, ens1), C.Var (u2, ens2)
    | C.Const (u1, ens1), C.Const (u2, ens2) when (UriManager.eq u1 u2) ->
        aux_ens table ens1 ens2
    | C.Cast (s1, t1), C.Cast (s2, t2)
    | C.Prod (_, s1, t1), C.Prod (_, s2, t2)
    | C.Lambda (_, s1, t1), C.Lambda (_, s2, t2)
    | C.LetIn (_, s1, t1), C.LetIn (_, s2, t2) ->
        let table = aux table s1 s2 in
        aux table t1 t2
    | C.Appl l1, C.Appl l2 -> (
        try List.fold_left2 (fun res t1 t2 -> (aux res t1 t2)) table l1 l2
        with Invalid_argument _ -> raise NotMetaConvertible
      )
    | C.MutInd (u1, i1, ens1), C.MutInd (u2, i2, ens2)
        when (UriManager.eq u1 u2) && i1 = i2 -> aux_ens table ens1 ens2
    | C.MutConstruct (u1, i1, j1, ens1), C.MutConstruct (u2, i2, j2, ens2)
        when (UriManager.eq u1 u2) && i1 = i2 && j1 = j2 ->
        aux_ens table ens1 ens2
    | C.MutCase (u1, i1, s1, t1, l1), C.MutCase (u2, i2, s2, t2, l2)
        when (UriManager.eq u1 u2) && i1 = i2 ->
        let table = aux table s1 s2 in
        let table = aux table t1 t2 in (
          try List.fold_left2 (fun res t1 t2 -> (aux res t1 t2)) table l1 l2
          with Invalid_argument _ -> raise NotMetaConvertible
        )
    | C.Fix (i1, il1), C.Fix (i2, il2) when i1 = i2 -> (
        try
          List.fold_left2
            (fun res (n1, i1, s1, t1) (n2, i2, s2, t2) ->
               if i1 <> i2 then raise NotMetaConvertible
               else
                 let res = (aux res s1 s2) in aux res t1 t2)
            table il1 il2
        with Invalid_argument _ -> raise NotMetaConvertible
      )
    | C.CoFix (i1, il1), C.CoFix (i2, il2) when i1 = i2 -> (
        try
          List.fold_left2
            (fun res (n1, s1, t1) (n2, s2, t2) ->
               let res = aux res s1 s2 in aux res t1 t2)
            table il1 il2
        with Invalid_argument _ -> raise NotMetaConvertible
      )
    | t1, t2 when t1 = t2 -> table
    | _, _ -> raise NotMetaConvertible
        
  and aux_ens table ens1 ens2 =
    let cmp (u1, t1) (u2, t2) =
      Pervasives.compare (UriManager.string_of_uri u1) (UriManager.string_of_uri u2)
    in
    let ens1 = List.sort cmp ens1
    and ens2 = List.sort cmp ens2 in
    try
      List.fold_left2
        (fun res (u1, t1) (u2, t2) ->
           if not (UriManager.eq u1 u2) then raise NotMetaConvertible
           else aux res t1 t2)
        table ens1 ens2
    with Invalid_argument _ -> raise NotMetaConvertible
  in
  aux table t1 t2
;;


let meta_convertibility_eq eq1 eq2 =
  let _, _, (ty, left, right, _), _,_ = open_equality eq1 in
  let _, _, (ty', left', right', _), _,_ = open_equality eq2 in
  if ty <> ty' then
    false
  else if (left = left') && (right = right') then
    true
  else if (left = right') && (right = left') then
    true
  else
    try
      let table = meta_convertibility_aux ([], []) left left' in
      let _ = meta_convertibility_aux table right right' in
      true
    with NotMetaConvertible ->
      try
        let table = meta_convertibility_aux ([], []) left right' in
        let _ = meta_convertibility_aux table right left' in
        true
      with NotMetaConvertible ->
        false
;;


let meta_convertibility t1 t2 =
  if t1 = t2 then
    true
  else
    try
      ignore(meta_convertibility_aux ([], []) t1 t2);
      true
    with NotMetaConvertible ->
      false
;;

exception TermIsNotAnEquality;;

let term_is_equality term =
  let iseq uri = UriManager.eq uri (LibraryObjects.eq_URI ()) in
  match term with
  | Cic.Appl [Cic.MutInd (uri, _, _); _; _; _] when iseq uri -> true
  | _ -> false
;;

let equality_of_term proof term =
  let eq_uri = LibraryObjects.eq_URI () in
  let iseq uri = UriManager.eq uri eq_uri in
  match term with
  | Cic.Appl [Cic.MutInd (uri, _, _); ty; t1; t2] when iseq uri ->
      let o = !Utils.compare_terms t1 t2 in
      let stat = (ty,t1,t2,o) in
      let w = Utils.compute_equality_weight stat in
      let e = mk_equality (w, (Exact proof, BasicProof ([],proof)),stat,[]) in
      e
  | _ ->
      raise TermIsNotAnEquality
;;

let is_weak_identity eq = 
  let _,_,(_,left, right,_),_,_ = open_equality eq in
  left = right || meta_convertibility left right 
;;

let is_identity (_, context, ugraph) eq = 
  let _,_,(ty,left,right,_),menv,_ = open_equality eq in
  left = right ||
  (* (meta_convertibility left right)) *)
  fst (CicReduction.are_convertible ~metasenv:menv context left right ugraph)
;;


let term_of_equality equality =
  let _, _, (ty, left, right, _), menv, _= open_equality equality in
  let eq i = function Cic.Meta (j, _) -> i = j | _ -> false in
  let argsno = List.length menv in
  let t =
    CicSubstitution.lift argsno
      (Cic.Appl [Cic.MutInd (LibraryObjects.eq_URI (), 0, []); ty; left; right])
  in
  snd (
    List.fold_right
      (fun (i,_,ty) (n, t) ->
         let name = Cic.Name ("X" ^ (string_of_int n)) in
         let ty = CicSubstitution.lift (n-1) ty in
         let t = 
           ProofEngineReduction.replace
             ~equality:eq ~what:[i]
             ~with_what:[Cic.Rel (argsno - (n - 1))] ~where:t
         in
           (n-1, Cic.Prod (name, ty, t)))
      menv (argsno, t))
;;