(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(* ML part of the Ring tactic *)

open Pp
open Util
open Flags
open Term
open Names
open Libnames
open Nameops
open Reductionops
open Tacticals
open Tacexpr
open Tacmach
open Printer
open Equality
open Vernacinterp
open Vernacexpr
open Libobject
open Closure
open Tacred
open Tactics
open Pattern
open Hiddentac
open Nametab
open Quote
open Mod_subst

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

let ring_dir = ["Coq";"ring"]
let setoids_dir = ["Coq";"Setoids"]

let ring_constant = Coqlib.gen_constant_in_modules "Ring"
  [ring_dir@["LegacyRing_theory"];
   ring_dir@["Setoid_ring_theory"];
   ring_dir@["Ring_normalize"];
   ring_dir@["Ring_abstract"];
   setoids_dir@["Setoid"];
   ring_dir@["Setoid_ring_normalize"]]

(* Ring theory *)
let coq_Ring_Theory = lazy (ring_constant "Ring_Theory")
let coq_Semi_Ring_Theory = lazy (ring_constant "Semi_Ring_Theory")

(* Setoid ring theory *)
let coq_Setoid_Ring_Theory = lazy (ring_constant "Setoid_Ring_Theory")
let coq_Semi_Setoid_Ring_Theory = lazy(ring_constant "Semi_Setoid_Ring_Theory")

(* Ring normalize *)
let coq_SPplus = lazy (ring_constant "SPplus")
let coq_SPmult = lazy (ring_constant "SPmult")
let coq_SPvar = lazy (ring_constant "SPvar")
let coq_SPconst = lazy (ring_constant "SPconst")
let coq_Pplus = lazy (ring_constant "Pplus")
let coq_Pmult = lazy (ring_constant "Pmult")
let coq_Pvar = lazy (ring_constant "Pvar")
let coq_Pconst = lazy (ring_constant "Pconst")
let coq_Popp = lazy (ring_constant "Popp")
let coq_interp_sp = lazy (ring_constant "interp_sp")
let coq_interp_p = lazy (ring_constant "interp_p")
let coq_interp_cs = lazy (ring_constant "interp_cs")
let coq_spolynomial_simplify = lazy (ring_constant "spolynomial_simplify")
let coq_polynomial_simplify = lazy (ring_constant "polynomial_simplify")
let coq_spolynomial_simplify_ok = lazy(ring_constant "spolynomial_simplify_ok")
let coq_polynomial_simplify_ok = lazy (ring_constant "polynomial_simplify_ok")

(* Setoid theory *)
let coq_Setoid_Theory = lazy(ring_constant "Setoid_Theory")

let coq_seq_refl = lazy(ring_constant "Seq_refl")
let coq_seq_sym = lazy(ring_constant "Seq_sym")
let coq_seq_trans = lazy(ring_constant "Seq_trans")

(* Setoid Ring normalize *)
let coq_SetSPplus = lazy (ring_constant "SetSPplus")
let coq_SetSPmult = lazy (ring_constant "SetSPmult")
let coq_SetSPvar = lazy (ring_constant "SetSPvar")
let coq_SetSPconst = lazy (ring_constant "SetSPconst")
let coq_SetPplus = lazy (ring_constant "SetPplus")
let coq_SetPmult = lazy (ring_constant "SetPmult")
let coq_SetPvar = lazy (ring_constant "SetPvar")
let coq_SetPconst = lazy (ring_constant "SetPconst")
let coq_SetPopp = lazy (ring_constant "SetPopp")
let coq_interp_setsp = lazy (ring_constant "interp_setsp")
let coq_interp_setp = lazy (ring_constant "interp_setp")
let coq_interp_setcs = lazy (ring_constant "interp_setcs")
let coq_setspolynomial_simplify =
  lazy (ring_constant "setspolynomial_simplify")
let coq_setpolynomial_simplify =
  lazy (ring_constant "setpolynomial_simplify")
let coq_setspolynomial_simplify_ok =
  lazy (ring_constant "setspolynomial_simplify_ok")
let coq_setpolynomial_simplify_ok =
  lazy (ring_constant "setpolynomial_simplify_ok")

(* Ring abstract *)
let coq_ASPplus = lazy (ring_constant "ASPplus")
let coq_ASPmult = lazy (ring_constant "ASPmult")
let coq_ASPvar = lazy (ring_constant "ASPvar")
let coq_ASP0 = lazy (ring_constant "ASP0")
let coq_ASP1 = lazy (ring_constant "ASP1")
let coq_APplus = lazy (ring_constant "APplus")
let coq_APmult = lazy (ring_constant "APmult")
let coq_APvar = lazy (ring_constant "APvar")
let coq_AP0 = lazy (ring_constant "AP0")
let coq_AP1 = lazy (ring_constant "AP1")
let coq_APopp = lazy (ring_constant "APopp")
let coq_interp_asp = lazy (ring_constant "interp_asp")
let coq_interp_ap = lazy (ring_constant "interp_ap")
let coq_interp_acs = lazy (ring_constant "interp_acs")
let coq_interp_sacs = lazy (ring_constant "interp_sacs")
let coq_aspolynomial_normalize = lazy (ring_constant "aspolynomial_normalize")
let coq_apolynomial_normalize = lazy (ring_constant "apolynomial_normalize")
let coq_aspolynomial_normalize_ok =
  lazy (ring_constant "aspolynomial_normalize_ok")
let coq_apolynomial_normalize_ok =
  lazy (ring_constant "apolynomial_normalize_ok")

(* Logic --> to be found in Coqlib *)
open Coqlib

let mkLApp(fc,v) = mkApp(Lazy.force fc, v)

(*********** Useful types and functions ************)

module OperSet =
  Set.Make (struct
	      type t = global_reference
	      let compare = (RefOrdered.compare : t->t->int)
	    end)

type morph =
    { plusm : constr;
      multm : constr;
      oppm : constr option;
    }

type theory =
    { th_ring : bool;                  (* false for a semi-ring *)
      th_abstract : bool;
      th_setoid : bool;                (* true for a setoid ring *)
      th_equiv : constr option;
      th_setoid_th : constr option;
      th_morph : morph option;
      th_a : constr;                   (* e.g. nat *)
      th_plus : constr;
      th_mult : constr;
      th_one : constr;
      th_zero : constr;
      th_opp : constr option;          (* None if semi-ring *)
      th_eq : constr;
      th_t : constr;                   (* e.g. NatTheory *)
      th_closed : ConstrSet.t;         (* e.g. [S; O] *)
                                       (* Must be empty for an abstract ring *)
    }

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

module Cmap = Map.Make(struct type t = constr let compare = constr_ord end)

let theories_map = ref Cmap.empty

let theories_map_add (c,t) = theories_map := Cmap.add c t !theories_map
let theories_map_find c = Cmap.find c !theories_map
let theories_map_mem c = Cmap.mem c !theories_map

let _ =
  Summary.declare_summary "tactic-ring-table"
    { Summary.freeze_function = (fun () -> !theories_map);
      Summary.unfreeze_function = (fun t -> theories_map := t);
      Summary.init_function = (fun () -> theories_map := Cmap.empty) }

(* declare a new type of object in the environment, "tactic-ring-theory"
   The functions theory_to_obj and obj_to_theory do the conversions
   between theories and environement objects. *)


let subst_morph subst morph =
  let plusm' = subst_mps subst morph.plusm in
  let multm' = subst_mps subst morph.multm in
  let oppm' = Option.smartmap (subst_mps subst) morph.oppm in
    if plusm' == morph.plusm
      && multm' == morph.multm
      && oppm' == morph.oppm then
	morph
    else
      { plusm = plusm' ;
	multm = multm' ;
	oppm = oppm' ;
      }

let subst_set subst cset =
  let same = ref true in
  let copy_subst c newset =
    let c' = subst_mps subst c in
      if not (c' == c) then same := false;
      ConstrSet.add c' newset
  in
  let cset' = ConstrSet.fold copy_subst cset ConstrSet.empty in
    if !same then cset else cset'

let subst_theory subst th =
  let th_equiv' = Option.smartmap (subst_mps subst) th.th_equiv in
  let th_setoid_th' = Option.smartmap (subst_mps subst) th.th_setoid_th in
  let th_morph' = Option.smartmap (subst_morph subst) th.th_morph in
  let th_a' = subst_mps subst th.th_a in
  let th_plus' = subst_mps subst th.th_plus in
  let th_mult' = subst_mps subst th.th_mult in
  let th_one' = subst_mps subst th.th_one in
  let th_zero' = subst_mps subst th.th_zero in
  let th_opp' = Option.smartmap (subst_mps subst) th.th_opp in
  let th_eq' = subst_mps subst th.th_eq in
  let th_t' = subst_mps subst th.th_t in
  let th_closed' = subst_set subst th.th_closed in
    if th_equiv' == th.th_equiv
      && th_setoid_th' == th.th_setoid_th
      && th_morph' == th.th_morph
      && th_a' == th.th_a
      && th_plus' == th.th_plus
      && th_mult' == th.th_mult
      && th_one' == th.th_one
      && th_zero' == th.th_zero
      && th_opp' == th.th_opp
      && th_eq' == th.th_eq
      && th_t' == th.th_t
      && th_closed' == th.th_closed
    then
      th
    else
    { th_ring = th.th_ring ;
      th_abstract = th.th_abstract ;
      th_setoid = th.th_setoid ;
      th_equiv = th_equiv' ;
      th_setoid_th = th_setoid_th' ;
      th_morph = th_morph' ;
      th_a = th_a' ;
      th_plus = th_plus' ;
      th_mult = th_mult' ;
      th_one = th_one' ;
      th_zero = th_zero' ;
      th_opp = th_opp' ;
      th_eq = th_eq' ;
      th_t = th_t' ;
      th_closed = th_closed' ;
    }


let subst_th (subst,(c,th as obj)) =
  let c' = subst_mps subst c in
  let th' = subst_theory subst th in
    if c' == c && th' == th then obj else
      (c',th')


let theory_to_obj : constr * theory -> obj =
  let cache_th (_,(c, th)) = theories_map_add (c,th) in
  declare_object {(default_object "tactic-ring-theory") with
		    open_function = (fun i o -> if i=1 then cache_th o);
                    cache_function = cache_th;
		    subst_function = subst_th;
		    classify_function = (fun x -> Substitute x) }

(* from the set A, guess the associated theory *)
(* With this simple solution, the theory to use is automatically guessed *)
(* But only one theory can be declared for a given Set *)

let guess_theory a =
  try
    theories_map_find a
  with Not_found ->
    errorlabstrm "Ring"
      (str "No Declared Ring Theory for " ++
	 pr_lconstr a ++ fnl () ++
	 str "Use Add [Semi] Ring to declare it")

(* Looks up an option *)

let unbox = function
  | Some w -> w
  | None -> anomaly "Ring : Not in case of a setoid ring."

(* Protects the convertibility test against undue exceptions when using it
   with untyped terms *)

let safe_pf_conv_x gl c1 c2 = try pf_conv_x gl c1 c2 with _ -> false


(* Add a Ring or a Semi-Ring to the database after a type verification *)

let implement_theory env t th args =
  is_conv env Evd.empty (Typing.type_of env Evd.empty t) (mkLApp (th, args))

(* (\* The following test checks whether the provided morphism is the default *)
(*    one for the given operation. In principle the test is too strict, since *)
(*    it should possible to provide another proof for the same fact (proof *)
(*    irrelevance). In particular, the error message is be not very explicative. *\) *)
let states_compatibility_for env plus mult opp morphs =
 let check op compat = true in
(*   is_conv env Evd.empty (Setoid_replace.default_morphism op).Setoid_replace.lem *)
(*    compat in *)
 check plus morphs.plusm &&
 check mult morphs.multm &&
 (match (opp,morphs.oppm) with
     None, None -> true
   | Some opp, Some compat -> check opp compat
   | _,_ -> assert false)

let add_theory want_ring want_abstract want_setoid a aequiv asetth amorph aplus amult aone azero aopp aeq t cset =
  if theories_map_mem a then errorlabstrm "Add Semi Ring"
    (str "A (Semi-)(Setoid-)Ring Structure is already declared for " ++
       pr_lconstr a);
  let env = Global.env () in
    if (want_ring & want_setoid & (
	not (implement_theory env t coq_Setoid_Ring_Theory
	  [| a; (unbox aequiv); aplus; amult; aone; azero; (unbox aopp); aeq|])
        ||
	not (implement_theory env (unbox asetth) coq_Setoid_Theory
	  [| a; (unbox aequiv) |]) ||
        not (states_compatibility_for env aplus amult aopp (unbox amorph))
        )) then
      errorlabstrm "addring" (str "Not a valid Setoid-Ring theory");
    if (not want_ring & want_setoid & (
        not (implement_theory env t coq_Semi_Setoid_Ring_Theory
	  [| a; (unbox aequiv); aplus; amult; aone; azero; aeq|]) ||
	not (implement_theory env (unbox asetth) coq_Setoid_Theory
	  [| a; (unbox aequiv) |]) ||
        not (states_compatibility_for env aplus amult aopp (unbox amorph))))
    then
      errorlabstrm "addring" (str "Not a valid Semi-Setoid-Ring theory");
    if (want_ring & not want_setoid &
	not (implement_theory env t coq_Ring_Theory
	  [| a; aplus; amult; aone; azero; (unbox aopp); aeq |])) then
      errorlabstrm "addring" (str "Not a valid Ring theory");
    if (not want_ring & not want_setoid &
	not (implement_theory env t coq_Semi_Ring_Theory
	  [| a; aplus; amult; aone; azero; aeq |])) then
      errorlabstrm "addring" (str "Not a valid Semi-Ring theory");
    Lib.add_anonymous_leaf
      (theory_to_obj
	 (a, { th_ring = want_ring;
	       th_abstract = want_abstract;
	       th_setoid = want_setoid;
	       th_equiv = aequiv;
	       th_setoid_th = asetth;
	       th_morph = amorph;
	       th_a = a;
	       th_plus = aplus;
	       th_mult = amult;
	       th_one = aone;
	       th_zero = azero;
	       th_opp = aopp;
	       th_eq = aeq;
	       th_t = t;
	       th_closed = cset }))

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

(*
   gl : goal sigma
   th : semi-ring theory (concrete)
   cl : constr list [c1; c2; ...]

Builds
   - a list of tuples [(c1, c'1, c''1, c'1_eq_c''1); ... ]
  where c'i is convertible with ci and
  c'i_eq_c''i is a proof of equality of c'i and c''i

*)

module Constrhash = Hashtbl.Make
  (struct type t = constr
	  let equal = eq_constr
	  let hash = hash_constr
   end)

let build_spolynom gl th lc =
  let varhash  = (Constrhash.create 17 : constr Constrhash.t) in
  let varlist = ref ([] : constr list) in (* list of variables *)
  let counter = ref 1 in (* number of variables created + 1 *)
  (* aux creates the spolynom p by a recursive destructuration of c
     and builds the varmap with side-effects *)
  let rec aux c =
    match (kind_of_term (strip_outer_cast c)) with
      | App (binop,[|c1; c2|]) when safe_pf_conv_x gl binop th.th_plus ->
	  mkLApp(coq_SPplus, [|th.th_a; aux c1; aux c2 |])
      | App (binop,[|c1; c2|]) when safe_pf_conv_x gl binop th.th_mult ->
	  mkLApp(coq_SPmult, [|th.th_a; aux c1; aux c2 |])
      | _ when closed_under th.th_closed c ->
	  mkLApp(coq_SPconst, [|th.th_a; c |])
      | _ ->
	  try Constrhash.find varhash c
	  with Not_found ->
	    let newvar =
              mkLApp(coq_SPvar, [|th.th_a; (path_of_int !counter) |]) in
	    begin
	      incr counter;
	      varlist := c :: !varlist;
	      Constrhash.add varhash c newvar;
	      newvar
	    end
  in
  let lp = List.map aux lc in
  let v = btree_of_array (Array.of_list (List.rev !varlist)) th.th_a in
  List.map
    (fun p ->
       (mkLApp (coq_interp_sp,
               [|th.th_a; th.th_plus; th.th_mult; th.th_zero; v; p |]),
	mkLApp (coq_interp_cs,
               [|th.th_a; th.th_plus; th.th_mult; th.th_one; th.th_zero; v;
		  pf_reduce cbv_betadeltaiota gl
		    (mkLApp (coq_spolynomial_simplify,
			    [| th.th_a; th.th_plus; th.th_mult;
			       th.th_one; th.th_zero;
			       th.th_eq; p|])) |]),
	mkLApp (coq_spolynomial_simplify_ok,
	        [| th.th_a; th.th_plus; th.th_mult; th.th_one; th.th_zero;
		  th.th_eq; v; th.th_t; p |])))
    lp

(*
   gl : goal sigma
   th : ring theory (concrete)
   cl : constr list [c1; c2; ...]

Builds
   - a list of tuples [(c1, c'1, c''1, c'1_eq_c''1); ... ]
  where c'i is convertible with ci and
  c'i_eq_c''i is a proof of equality of c'i and c''i

*)

let build_polynom gl th lc =
  let varhash  = (Constrhash.create 17 : constr Constrhash.t) in
  let varlist = ref ([] : constr list) in (* list of variables *)
  let counter = ref 1 in (* number of variables created + 1 *)
  let rec aux c =
    match (kind_of_term (strip_outer_cast c)) with
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_plus ->
	  mkLApp(coq_Pplus, [|th.th_a; aux c1; aux c2 |])
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_mult ->
	  mkLApp(coq_Pmult, [|th.th_a; aux c1; aux c2 |])
      (* The special case of Z.sub *)
      | App (binop, [|c1; c2|])
	  when safe_pf_conv_x gl c
            (mkApp (th.th_plus, [|c1; mkApp(unbox th.th_opp, [|c2|])|])) ->
	    mkLApp(coq_Pplus,
                  [|th.th_a; aux c1;
	            mkLApp(coq_Popp, [|th.th_a; aux c2|]) |])
      | App (unop, [|c1|]) when safe_pf_conv_x gl unop (unbox th.th_opp) ->
	  mkLApp(coq_Popp, [|th.th_a; aux c1|])
      | _ when closed_under th.th_closed c ->
	  mkLApp(coq_Pconst, [|th.th_a; c |])
      | _ ->
	  try Constrhash.find varhash c
	  with Not_found ->
	    let newvar =
              mkLApp(coq_Pvar, [|th.th_a; (path_of_int !counter) |]) in
	    begin
	      incr counter;
	      varlist := c :: !varlist;
	      Constrhash.add varhash c newvar;
	      newvar
	    end
  in
  let lp = List.map aux lc in
  let v = (btree_of_array (Array.of_list (List.rev !varlist)) th.th_a) in
  List.map
    (fun p ->
       (mkLApp(coq_interp_p,
	       [| th.th_a; th.th_plus; th.th_mult; th.th_zero;
                 (unbox th.th_opp); v; p |])),
	mkLApp(coq_interp_cs,
	       [| th.th_a; th.th_plus; th.th_mult; th.th_one; th.th_zero; v;
		  pf_reduce cbv_betadeltaiota gl
		    (mkLApp(coq_polynomial_simplify,
			    [| th.th_a; th.th_plus; th.th_mult;
			       th.th_one; th.th_zero;
			       (unbox th.th_opp); th.th_eq; p |])) |]),
	mkLApp(coq_polynomial_simplify_ok,
	       [| th.th_a; th.th_plus; th.th_mult; th.th_one; th.th_zero;
		  (unbox th.th_opp); th.th_eq; v; th.th_t; p |]))
    lp

(*
   gl : goal sigma
   th : semi-ring theory (abstract)
   cl : constr list [c1; c2; ...]

Builds
   - a list of tuples [(c1, c'1, c''1, c'1_eq_c''1); ... ]
  where c'i is convertible with ci and
  c'i_eq_c''i is a proof of equality of c'i and c''i

*)

let build_aspolynom gl th lc =
  let varhash  = (Constrhash.create 17 : constr Constrhash.t) in
  let varlist = ref ([] : constr list) in (* list of variables *)
  let counter = ref 1 in (* number of variables created + 1 *)
  (* aux creates the aspolynom p by a recursive destructuration of c
     and builds the varmap with side-effects *)
  let rec aux c =
    match (kind_of_term (strip_outer_cast c)) with
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_plus ->
	  mkLApp(coq_ASPplus, [| aux c1; aux c2 |])
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_mult ->
	  mkLApp(coq_ASPmult, [| aux c1; aux c2 |])
      | _ when safe_pf_conv_x gl c th.th_zero -> Lazy.force coq_ASP0
      | _ when safe_pf_conv_x gl c th.th_one -> Lazy.force coq_ASP1
      | _ ->
	  try Constrhash.find varhash c
	  with Not_found ->
	    let newvar = mkLApp(coq_ASPvar, [|(path_of_int !counter) |]) in
	    begin
	      incr counter;
	      varlist := c :: !varlist;
	      Constrhash.add varhash c newvar;
	      newvar
	    end
  in
  let lp = List.map aux lc in
  let v = btree_of_array (Array.of_list (List.rev !varlist)) th.th_a in
  List.map
    (fun p ->
       (mkLApp(coq_interp_asp,
               [| th.th_a; th.th_plus; th.th_mult;
		  th.th_one; th.th_zero; v; p |]),
	mkLApp(coq_interp_acs,
               [| th.th_a; th.th_plus; th.th_mult;
		  th.th_one; th.th_zero; v;
		  pf_reduce cbv_betadeltaiota gl
		    (mkLApp(coq_aspolynomial_normalize,[|p|])) |]),
	mkLApp(coq_spolynomial_simplify_ok,
	       [| th.th_a; th.th_plus; th.th_mult; th.th_one; th.th_zero;
		  th.th_eq; v; th.th_t; p |])))
    lp

(*
   gl : goal sigma
   th : ring theory (abstract)
   cl : constr list [c1; c2; ...]

Builds
   - a list of tuples [(c1, c'1, c''1, c'1_eq_c''1); ... ]
  where c'i is convertible with ci and
  c'i_eq_c''i is a proof of equality of c'i and c''i

*)

let build_apolynom gl th lc =
  let varhash  = (Constrhash.create 17 : constr Constrhash.t) in
  let varlist = ref ([] : constr list) in (* list of variables *)
  let counter = ref 1 in (* number of variables created + 1 *)
  let rec aux c =
    match (kind_of_term (strip_outer_cast c)) with
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_plus ->
	  mkLApp(coq_APplus, [| aux c1; aux c2 |])
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_mult ->
	  mkLApp(coq_APmult, [| aux c1; aux c2 |])
      (* The special case of Z.sub *)
      | App (binop, [|c1; c2|])
	  when safe_pf_conv_x gl c
            (mkApp(th.th_plus, [|c1; mkApp(unbox th.th_opp,[|c2|]) |])) ->
	    mkLApp(coq_APplus,
                   [|aux c1; mkLApp(coq_APopp,[|aux c2|]) |])
      | App (unop, [|c1|]) when safe_pf_conv_x gl unop (unbox th.th_opp) ->
	  mkLApp(coq_APopp, [| aux c1 |])
      | _ when safe_pf_conv_x gl c th.th_zero -> Lazy.force coq_AP0
      | _ when safe_pf_conv_x gl c th.th_one -> Lazy.force coq_AP1
      | _ ->
	  try Constrhash.find varhash c
	  with Not_found ->
	    let newvar =
              mkLApp(coq_APvar, [| path_of_int !counter |]) in
	    begin
	      incr counter;
	      varlist := c :: !varlist;
	      Constrhash.add varhash c newvar;
	      newvar
	    end
  in
  let lp = List.map aux lc in
  let v = (btree_of_array (Array.of_list (List.rev !varlist)) th.th_a) in
  List.map
    (fun p ->
       (mkLApp(coq_interp_ap,
	       [| th.th_a; th.th_plus; th.th_mult; th.th_one;
		  th.th_zero; (unbox th.th_opp); v; p |]),
	mkLApp(coq_interp_sacs,
	       [| th.th_a; th.th_plus; th.th_mult;
		  th.th_one; th.th_zero; (unbox th.th_opp); v;
		  pf_reduce cbv_betadeltaiota gl
		    (mkLApp(coq_apolynomial_normalize, [|p|])) |]),
	mkLApp(coq_apolynomial_normalize_ok,
	       [| th.th_a; th.th_plus; th.th_mult; th.th_one; th.th_zero;
		  (unbox th.th_opp); th.th_eq; v; th.th_t; p |])))
    lp

(*
   gl : goal sigma
   th : setoid ring theory (concrete)
   cl : constr list [c1; c2; ...]

Builds
   - a list of tuples [(c1, c'1, c''1, c'1_eq_c''1); ... ]
  where c'i is convertible with ci and
  c'i_eq_c''i is a proof of equality of c'i and c''i

*)

let build_setpolynom gl th lc =
  let varhash  = (Constrhash.create 17 : constr Constrhash.t) in
  let varlist = ref ([] : constr list) in (* list of variables *)
  let counter = ref 1 in (* number of variables created + 1 *)
  let rec aux c =
    match (kind_of_term (strip_outer_cast c)) with
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_plus ->
	  mkLApp(coq_SetPplus, [|th.th_a; aux c1; aux c2 |])
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_mult ->
	  mkLApp(coq_SetPmult, [|th.th_a; aux c1; aux c2 |])
      (* The special case of Z.sub *)
      | App (binop, [|c1; c2|])
	  when safe_pf_conv_x gl c
	    (mkApp(th.th_plus, [|c1; mkApp(unbox th.th_opp,[|c2|])|])) ->
	      mkLApp(coq_SetPplus,
                     [| th.th_a; aux c1;
			mkLApp(coq_SetPopp, [|th.th_a; aux c2|]) |])
      | App (unop, [|c1|]) when safe_pf_conv_x gl unop (unbox th.th_opp) ->
	  mkLApp(coq_SetPopp, [| th.th_a; aux c1 |])
      | _ when closed_under th.th_closed c ->
	  mkLApp(coq_SetPconst, [| th.th_a; c |])
      | _ ->
	  try Constrhash.find varhash c
	  with Not_found ->
	    let newvar =
              mkLApp(coq_SetPvar, [| th.th_a; path_of_int !counter |]) in
	    begin
	      incr counter;
	      varlist := c :: !varlist;
	      Constrhash.add varhash c newvar;
	      newvar
	    end
  in
  let lp = List.map aux lc in
  let v = (btree_of_array (Array.of_list (List.rev !varlist)) th.th_a) in
  List.map
    (fun p ->
       (mkLApp(coq_interp_setp,
	       [| th.th_a; th.th_plus; th.th_mult; th.th_zero;
                  (unbox th.th_opp); v; p |]),
	mkLApp(coq_interp_setcs,
	       [| th.th_a; th.th_plus; th.th_mult; th.th_one; th.th_zero; v;
		  pf_reduce cbv_betadeltaiota gl
		    (mkLApp(coq_setpolynomial_simplify,
			    [| th.th_a; th.th_plus; th.th_mult;
			       th.th_one; th.th_zero;
			       (unbox th.th_opp); th.th_eq; p |])) |]),
	mkLApp(coq_setpolynomial_simplify_ok,
	       [| th.th_a; (unbox th.th_equiv); th.th_plus;
                  th.th_mult; th.th_one; th.th_zero;(unbox th.th_opp);
                  th.th_eq; (unbox th.th_setoid_th);
		  (unbox th.th_morph).plusm; (unbox th.th_morph).multm;
		  (unbox (unbox th.th_morph).oppm); v; th.th_t; p |])))
    lp

(*
   gl : goal sigma
   th : semi setoid ring theory (concrete)
   cl : constr list [c1; c2; ...]

Builds
   - a list of tuples [(c1, c'1, c''1, c'1_eq_c''1); ... ]
  where c'i is convertible with ci and
  c'i_eq_c''i is a proof of equality of c'i and c''i

*)

let build_setspolynom gl th lc =
  let varhash  = (Constrhash.create 17 : constr Constrhash.t) in
  let varlist = ref ([] : constr list) in (* list of variables *)
  let counter = ref 1 in (* number of variables created + 1 *)
  let rec aux c =
    match (kind_of_term (strip_outer_cast c)) with
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_plus ->
	  mkLApp(coq_SetSPplus, [|th.th_a; aux c1; aux c2 |])
      | App (binop, [|c1; c2|]) when safe_pf_conv_x gl binop th.th_mult ->
	  mkLApp(coq_SetSPmult, [| th.th_a; aux c1; aux c2 |])
      | _ when closed_under th.th_closed c ->
	  mkLApp(coq_SetSPconst, [| th.th_a; c |])
      | _ ->
	  try Constrhash.find varhash c
	  with Not_found ->
	    let newvar =
              mkLApp(coq_SetSPvar, [|th.th_a; path_of_int !counter |]) in
	    begin
	      incr counter;
	      varlist := c :: !varlist;
	      Constrhash.add varhash c newvar;
	      newvar
	    end
  in
  let lp = List.map aux lc in
  let v = (btree_of_array (Array.of_list (List.rev !varlist)) th.th_a) in
  List.map
    (fun p ->
       (mkLApp(coq_interp_setsp,
	       [| th.th_a; th.th_plus; th.th_mult; th.th_zero; v; p |]),
	mkLApp(coq_interp_setcs,
               [| th.th_a; th.th_plus; th.th_mult; th.th_one; th.th_zero; v;
		  pf_reduce cbv_betadeltaiota gl
		    (mkLApp(coq_setspolynomial_simplify,
			    [| th.th_a; th.th_plus; th.th_mult;
			       th.th_one; th.th_zero;
			       th.th_eq; p |])) |]),
	mkLApp(coq_setspolynomial_simplify_ok,
	       [| th.th_a; (unbox th.th_equiv); th.th_plus;
                  th.th_mult; th.th_one; th.th_zero; th.th_eq;
                  (unbox th.th_setoid_th);
		  (unbox th.th_morph).plusm;
                  (unbox th.th_morph).multm; v; th.th_t; p |])))
    lp

module SectionPathSet =
  Set.Make(struct
	     type t = full_path
	     let compare = Pervasives.compare
	   end)

(* Avec l'uniformisation des red_kind, on perd ici sur la structure
   SectionPathSet; peut-être faudra-t-il la déplacer dans Closure *)
let constants_to_unfold =
(*  List.fold_right SectionPathSet.add *)
  let transform s =
    let sp = path_of_string s in
    let dir, id = repr_path sp in
      Libnames.encode_con dir id
  in
  List.map transform
    [ "Coq.ring.Ring_normalize.interp_cs";
      "Coq.ring.Ring_normalize.interp_var";
      "Coq.ring.Ring_normalize.interp_vl";
      "Coq.ring.Ring_abstract.interp_acs";
      "Coq.ring.Ring_abstract.interp_sacs";
      "Coq.quote.Quote.varmap_find";
      (* anciennement des Local devenus Definition *)
      "Coq.ring.Ring_normalize.ics_aux";
      "Coq.ring.Ring_normalize.ivl_aux";
      "Coq.ring.Ring_normalize.interp_m";
      "Coq.ring.Ring_abstract.iacs_aux";
      "Coq.ring.Ring_abstract.isacs_aux";
      "Coq.ring.Setoid_ring_normalize.interp_cs";
      "Coq.ring.Setoid_ring_normalize.interp_var";
      "Coq.ring.Setoid_ring_normalize.interp_vl";
      "Coq.ring.Setoid_ring_normalize.ics_aux";
      "Coq.ring.Setoid_ring_normalize.ivl_aux";
      "Coq.ring.Setoid_ring_normalize.interp_m";
    ]
(*    SectionPathSet.empty *)

(* Unfolds the functions interp and find_btree in the term c of goal gl *)
open RedFlags
let polynom_unfold_tac =
  let flags =
    (mkflags(fBETA::fIOTA::(List.map fCONST constants_to_unfold))) in
  reduct_in_concl (cbv_norm_flags flags,DEFAULTcast)

let polynom_unfold_tac_in_term gl =
  let flags =
    (mkflags(fBETA::fIOTA::fZETA::(List.map fCONST constants_to_unfold)))
  in
  cbv_norm_flags flags (pf_env gl) (project gl)

(* lc : constr list *)
(* th : theory associated to t *)
(* op : clause (None for conclusion or Some id for hypothesis id) *)
(* gl : goal  *)
(* Does the rewriting c_i -> (interp R RC v (polynomial_simplify p_i))
   where the ring R, the Ring theory RC, the varmap v and the polynomials p_i
   are guessed and such that c_i = (interp R RC v p_i) *)
let raw_polynom th op lc gl =
  (* first we sort the terms : if t' is a subterm of t it must appear
     after t in the list. This is to avoid that the normalization of t'
     modifies t in a non-desired way *)
  let lc = sort_subterm gl lc in
  let ltriplets =
    if th.th_setoid then
      if th.th_ring
      then build_setpolynom gl th lc
      else build_setspolynom gl th lc
    else
      if th.th_ring then
	if th.th_abstract
	then build_apolynom gl th lc
	else build_polynom gl th lc
      else
	if th.th_abstract
	then build_aspolynom gl th lc
	else build_spolynom gl th lc in
  let polynom_tac =
    List.fold_right2
      (fun ci (c'i, c''i, c'i_eq_c''i) tac ->
         let c'''i =
	   if !term_quality then polynom_unfold_tac_in_term gl c''i else c''i
	 in
         if !term_quality && safe_pf_conv_x gl c'''i ci then
	   tac (* convertible terms *)
         else if th.th_setoid
	 then
           (tclORELSE
              (tclORELSE
		 (h_exact c'i_eq_c''i)
		 (h_exact (mkLApp(coq_seq_sym,
				  [| th.th_a; (unbox th.th_equiv);
                                     (unbox th.th_setoid_th);
				     c'''i; ci; c'i_eq_c''i |]))))
	      (tclTHENS
		 (tclORELSE
                   (Equality.general_rewrite true
		     Termops.all_occurrences true false c'i_eq_c''i)
                   (Equality.general_rewrite false
		     Termops.all_occurrences true false c'i_eq_c''i))
                 [tac]))
	 else
           (tclORELSE
              (tclORELSE
		 (h_exact c'i_eq_c''i)
		 (h_exact (mkApp(build_coq_eq_sym (),
				 [|th.th_a; c'''i; ci; c'i_eq_c''i |]))))
	      (tclTHENS
		 (elim_type
		    (mkApp(build_coq_eq (), [|th.th_a; c'''i; ci |])))
		 [ tac;
                   h_exact c'i_eq_c''i ]))
)
      lc ltriplets polynom_unfold_tac
  in
  polynom_tac gl

let guess_eq_tac th =
  (tclORELSE reflexivity
     (tclTHEN
	polynom_unfold_tac
        (tclTHEN
	   (* Normalized sums associate on the right *)
	   (tclREPEAT
	      (tclTHENFIRST
		 (apply (mkApp(build_coq_f_equal2 (),
		               [| th.th_a; th.th_a; th.th_a;
				  th.th_plus |])))
		 reflexivity))
	   (tclTRY
	      (tclTHENLAST
		 (apply (mkApp(build_coq_f_equal2 (),
			       [| th.th_a; th.th_a; th.th_a;
				  th.th_plus |])))
		 reflexivity)))))

let guess_equiv_tac th =
  (tclORELSE (apply (mkLApp(coq_seq_refl,
			    [| th.th_a; (unbox th.th_equiv);
			       (unbox th.th_setoid_th)|])))
     (tclTHEN
	polynom_unfold_tac
	(tclREPEAT
	   (tclORELSE
	      (apply (unbox th.th_morph).plusm)
	      (apply (unbox th.th_morph).multm)))))

let match_with_equiv c = match (kind_of_term c) with
  | App (e,a) ->
      if (List.mem e []) (*  (Setoid_replace.equiv_list ())) *)
      then Some (decompose_app c)
      else None
  | _ -> None

let polynom lc gl =
  Coqlib.check_required_library ["Coq";"ring";"LegacyRing"];
  match lc with
   (* If no argument is given, try to recognize either an equality or
      a declared relation with arguments c1 ... cn,
      do "Ring c1 c2 ... cn" and then try to apply the simplification
      theorems declared for the relation *)
    | [] ->
	(try
	  match Hipattern.match_with_equation (pf_concl gl) with
	   | _,_,Hipattern.PolymorphicLeibnizEq (t,c1,c2) ->
	       let th = guess_theory t in
	       (tclTHEN (raw_polynom th None [c1;c2]) (guess_eq_tac th)) gl
	   | _,_,Hipattern.HeterogenousEq (t1,c1,t2,c2)
	       when safe_pf_conv_x gl t1 t2 ->
	       let th = guess_theory t1 in
	       (tclTHEN (raw_polynom th None [c1;c2]) (guess_eq_tac th)) gl
	   | _ -> raise Exit
	  with Hipattern.NoEquationFound | Exit ->
	          (match match_with_equiv (pf_concl gl) with
		     | Some (equiv, c1::args) ->
			 let t = (pf_type_of gl c1) in
			 let th = (guess_theory t) in
			   if List.exists
			     (fun c2 -> not (safe_pf_conv_x gl t (pf_type_of gl c2))) args
			   then
			     errorlabstrm "Ring :"
			       (str" All terms must have the same type");
			   (tclTHEN (raw_polynom th None (c1::args)) (guess_equiv_tac th)) gl
		     | _ -> errorlabstrm "polynom :"
			 (str" This goal is not an equality nor a setoid equivalence")))
    (* Elsewhere, guess the theory, check that all terms have the same type
       and apply raw_polynom *)
    | c :: lc' ->
	let t = pf_type_of gl c in
	let th = guess_theory t in
	  if List.exists
	    (fun c1 -> not (safe_pf_conv_x gl t (pf_type_of gl c1))) lc'
	  then
	    errorlabstrm "Ring :"
	      (str" All terms must have the same type");
	  (tclTHEN (raw_polynom th None lc) polynom_unfold_tac) gl