(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)



(* La tactique Fourier ne fonctionne de manière sûre que si les coefficients
des inéquations et équations sont entiers. En attendant la tactique Field.
*)

open Term
open Tactics
open Clenv
open Names
open Libnames
open Tacticals
open Tacmach
open Fourier
open Contradiction

(******************************************************************************
Opérations sur les combinaisons linéaires affines.
La partie homogène d'une combinaison linéaire est en fait une table de hash
qui donne le coefficient d'un terme du calcul des constructions,
qui est zéro si le terme n'y est pas.
*)

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

type flin = {fhom: rational Constrhash.t;
             fcste:rational};;

let flin_zero () = {fhom=Constrhash.create 50;fcste=r0};;

let flin_coef f x = try (Constrhash.find f.fhom x) with Not_found -> r0;;

let flin_add f x c =
    let cx = flin_coef f x in
    Constrhash.remove f.fhom x;
    Constrhash.add f.fhom x (rplus cx c);
    f
;;
let flin_add_cste f c =
    {fhom=f.fhom;
     fcste=rplus f.fcste c}
;;

let flin_one () = flin_add_cste (flin_zero()) r1;;

let flin_plus f1 f2 =
    let f3 = flin_zero() in
    Constrhash.iter (fun x c -> let _=flin_add f3 x c in ()) f1.fhom;
    Constrhash.iter (fun x c -> let _=flin_add f3 x c in ()) f2.fhom;
    flin_add_cste (flin_add_cste f3 f1.fcste) f2.fcste;
;;

let flin_minus f1 f2 =
    let f3 = flin_zero() in
    Constrhash.iter (fun x c -> let _=flin_add f3 x c in ()) f1.fhom;
    Constrhash.iter (fun x c -> let _=flin_add f3 x (rop c) in ()) f2.fhom;
    flin_add_cste (flin_add_cste f3 f1.fcste) (rop f2.fcste);
;;
let flin_emult a f =
    let f2 = flin_zero() in
    Constrhash.iter (fun x c -> let _=flin_add f2 x (rmult a c) in ()) f.fhom;
    flin_add_cste f2 (rmult a f.fcste);
;;

(*****************************************************************************)
open Vernacexpr

type ineq = Rlt | Rle | Rgt | Rge

let string_of_R_constant kn =
  match Names.repr_con kn with
    | MPfile dir, sec_dir, id when
	sec_dir = empty_dirpath &&
	string_of_dirpath dir = "Coq.Reals.Rdefinitions"
	-> string_of_label id
    | _ -> "constant_not_of_R"

let rec string_of_R_constr c =
 match kind_of_term c with
   Cast (c,_,_) -> string_of_R_constr c
  |Const c -> string_of_R_constant c
  | _ -> "not_of_constant"

let rec rational_of_constr c =
  match kind_of_term c with
  | Cast (c,_,_) -> (rational_of_constr c)
  | App (c,args) ->
      (match (string_of_R_constr c) with
	 | "Ropp" ->
	     rop (rational_of_constr args.(0))
	 | "Rinv" ->
	     rinv (rational_of_constr args.(0))
	 | "Rmult" ->
	     rmult (rational_of_constr args.(0))
                   (rational_of_constr args.(1))
	 | "Rdiv" ->
	     rdiv (rational_of_constr args.(0))
                  (rational_of_constr args.(1))
	 | "Rplus" ->
	     rplus (rational_of_constr args.(0))
                   (rational_of_constr args.(1))
	 | "Rminus" ->
	     rminus (rational_of_constr args.(0))
                    (rational_of_constr args.(1))
	 | _ -> failwith "not a rational")
  | Const kn ->
      (match (string_of_R_constant kn) with
	       "R1" -> r1
              |"R0" -> r0
              |  _ -> failwith "not a rational")
  |  _ -> failwith "not a rational"
;;

let rec flin_of_constr c =
  try(
    match kind_of_term c with
  | Cast (c,_,_) -> (flin_of_constr c)
  | App (c,args) ->
      (match (string_of_R_constr c) with
	   "Ropp" ->
             flin_emult (rop r1) (flin_of_constr args.(0))
	 | "Rplus"->
             flin_plus (flin_of_constr args.(0))
	               (flin_of_constr args.(1))
	 | "Rminus"->
             flin_minus (flin_of_constr args.(0))
	                (flin_of_constr args.(1))
	 | "Rmult"->
	     (try (let a=(rational_of_constr args.(0)) in
                     try (let b = (rational_of_constr args.(1)) in
			    (flin_add_cste (flin_zero()) (rmult a b)))
		     with e when Errors.noncritical e ->
                       (flin_add (flin_zero())
				 args.(1)
				 a))
	      with e when Errors.noncritical e ->
                (flin_add (flin_zero())
			  args.(0)
			  (rational_of_constr args.(1))))
	 | "Rinv"->
	     let a=(rational_of_constr args.(0)) in
	       flin_add_cste (flin_zero()) (rinv a)
	 | "Rdiv"->
	     (let b=(rational_of_constr args.(1)) in
		try (let a = (rational_of_constr args.(0)) in
		       (flin_add_cste (flin_zero()) (rdiv a b)))
		with e when Errors.noncritical e ->
                  (flin_add (flin_zero())
		            args.(0)
                            (rinv b)))
         |_->assert false)
  | Const c ->
        (match (string_of_R_constant c) with
	       "R1" -> flin_one ()
              |"R0" -> flin_zero ()
              |_-> assert false)
  |_-> assert false)
  with e when Errors.noncritical e ->
    flin_add (flin_zero())
                     c
	             r1
;;

let flin_to_alist f =
    let res=ref [] in
    Constrhash.iter (fun x c -> res:=(c,x)::(!res)) f;
    !res
;;

(* Représentation des hypothèses qui sont des inéquations ou des équations.
*)
type hineq={hname:constr; (* le nom de l'hypothèse *)
            htype:string; (* Rlt, Rgt, Rle, Rge, eqTLR ou eqTRL *)
            hleft:constr;
            hright:constr;
            hflin:flin;
            hstrict:bool}
;;

(* Transforme une hypothese h:t en inéquation flin<0 ou flin<=0
*)
let ineq1_of_constr (h,t) =
    match (kind_of_term t) with
       App (f,args) ->
         (match kind_of_term f with
           Const c when Array.length args = 2 ->
             let t1= args.(0) in
             let t2= args.(1) in
            (match (string_of_R_constant c) with
		 "Rlt" -> [{hname=h;
                           htype="Rlt";
		           hleft=t1;
			   hright=t2;
			   hflin= flin_minus (flin_of_constr t1)
                                             (flin_of_constr t2);
			   hstrict=true}]
		|"Rgt" -> [{hname=h;
                           htype="Rgt";
		           hleft=t2;
			   hright=t1;
			   hflin= flin_minus (flin_of_constr t2)
                                             (flin_of_constr t1);
			   hstrict=true}]
		|"Rle" -> [{hname=h;
                           htype="Rle";
		           hleft=t1;
			   hright=t2;
			   hflin= flin_minus (flin_of_constr t1)
                                             (flin_of_constr t2);
			   hstrict=false}]
		|"Rge" -> [{hname=h;
                           htype="Rge";
		           hleft=t2;
			   hright=t1;
			   hflin= flin_minus (flin_of_constr t2)
                                             (flin_of_constr t1);
			   hstrict=false}]
                |_->assert false)
          | Ind (kn,i) ->
	      if IndRef(kn,i) = Coqlib.glob_eq then
		           let t0= args.(0) in
                           let t1= args.(1) in
                           let t2= args.(2) in
		    (match (kind_of_term t0) with
                         Const c ->
			   (match (string_of_R_constant c) with
			      "R"->
                         [{hname=h;
                           htype="eqTLR";
		           hleft=t1;
			   hright=t2;
			   hflin= flin_minus (flin_of_constr t1)
                                             (flin_of_constr t2);
			   hstrict=false};
                          {hname=h;
                           htype="eqTRL";
		           hleft=t2;
			   hright=t1;
			   hflin= flin_minus (flin_of_constr t2)
                                             (flin_of_constr t1);
			   hstrict=false}]
                           |_-> assert false)
                         |_-> assert false)
	      else
		assert false
          |_-> assert false)
        |_-> assert false
;;

(* Applique la méthode de Fourier à une liste d'hypothèses (type hineq)
*)

let fourier_lineq lineq1 =
   let nvar=ref (-1) in
   let hvar=Constrhash.create 50 in (* la table des variables des inéquations *)
   List.iter (fun f ->
		Constrhash.iter (fun x _ -> if not (Constrhash.mem hvar x) then begin
				nvar:=(!nvar)+1;
				Constrhash.add hvar x (!nvar)
			      end)
                  f.hflin.fhom)
             lineq1;
   let sys= List.map (fun h->
               let v=Array.create ((!nvar)+1) r0 in
               Constrhash.iter (fun x c -> v.(Constrhash.find hvar x)<-c)
                  h.hflin.fhom;
               ((Array.to_list v)@[rop h.hflin.fcste],h.hstrict))
             lineq1 in
   unsolvable sys
;;

(*********************************************************************)
(* Defined constants *)

let get = Lazy.force
let constant = Coqlib.gen_constant "Fourier"

(* Standard library *)
open Coqlib
let coq_sym_eqT = lazy (build_coq_eq_sym ())
let coq_False = lazy (build_coq_False ())
let coq_not = lazy (build_coq_not ())
let coq_eq = lazy (build_coq_eq ())

(* Rdefinitions *)
let constant_real = constant ["Reals";"Rdefinitions"]

let coq_Rlt = lazy (constant_real "Rlt")
let coq_Rgt = lazy (constant_real "Rgt")
let coq_Rle = lazy (constant_real "Rle")
let coq_Rge = lazy (constant_real "Rge")
let coq_R = lazy (constant_real "R")
let coq_Rminus = lazy (constant_real "Rminus")
let coq_Rmult = lazy (constant_real "Rmult")
let coq_Rplus = lazy (constant_real "Rplus")
let coq_Ropp = lazy (constant_real "Ropp")
let coq_Rinv = lazy (constant_real "Rinv")
let coq_R0 = lazy (constant_real "R0")
let coq_R1 = lazy (constant_real "R1")

(* RIneq *)
let coq_Rinv_1 = lazy (constant ["Reals";"RIneq"] "Rinv_1")

(* Fourier_util *)
let constant_fourier = constant ["fourier";"Fourier_util"]

let coq_Rlt_zero_1 = lazy (constant_fourier "Rlt_zero_1")
let coq_Rlt_zero_pos_plus1 = lazy (constant_fourier "Rlt_zero_pos_plus1")
let coq_Rle_zero_pos_plus1 = lazy (constant_fourier "Rle_zero_pos_plus1")
let coq_Rlt_mult_inv_pos = lazy (constant_fourier "Rlt_mult_inv_pos")
let coq_Rle_zero_zero = lazy (constant_fourier "Rle_zero_zero")
let coq_Rle_zero_1 = lazy (constant_fourier "Rle_zero_1")
let coq_Rle_mult_inv_pos = lazy (constant_fourier "Rle_mult_inv_pos")
let coq_Rnot_lt0 = lazy (constant_fourier "Rnot_lt0")
let coq_Rle_not_lt = lazy (constant_fourier "Rle_not_lt")
let coq_Rfourier_gt_to_lt = lazy (constant_fourier "Rfourier_gt_to_lt")
let coq_Rfourier_ge_to_le = lazy (constant_fourier "Rfourier_ge_to_le")
let coq_Rfourier_eqLR_to_le = lazy (constant_fourier "Rfourier_eqLR_to_le")
let coq_Rfourier_eqRL_to_le = lazy (constant_fourier "Rfourier_eqRL_to_le")

let coq_Rfourier_not_ge_lt = lazy (constant_fourier "Rfourier_not_ge_lt")
let coq_Rfourier_not_gt_le = lazy (constant_fourier "Rfourier_not_gt_le")
let coq_Rfourier_not_le_gt = lazy (constant_fourier "Rfourier_not_le_gt")
let coq_Rfourier_not_lt_ge = lazy (constant_fourier "Rfourier_not_lt_ge")
let coq_Rfourier_lt = lazy (constant_fourier "Rfourier_lt")
let coq_Rfourier_le = lazy (constant_fourier "Rfourier_le")
let coq_Rfourier_lt_lt = lazy (constant_fourier "Rfourier_lt_lt")
let coq_Rfourier_lt_le = lazy (constant_fourier "Rfourier_lt_le")
let coq_Rfourier_le_lt = lazy (constant_fourier "Rfourier_le_lt")
let coq_Rfourier_le_le = lazy (constant_fourier "Rfourier_le_le")
let coq_Rnot_lt_lt = lazy (constant_fourier "Rnot_lt_lt")
let coq_Rnot_le_le = lazy (constant_fourier "Rnot_le_le")
let coq_Rlt_not_le_frac_opp = lazy (constant_fourier "Rlt_not_le_frac_opp")

(******************************************************************************
Construction de la preuve en cas de succès de la méthode de Fourier,
i.e. on obtient une contradiction.
*)
let is_int x = (x.den)=1
;;

(* fraction = couple (num,den) *)
let rec rational_to_fraction x= (x.num,x.den)
;;

(* traduction -3 -> (Ropp (Rplus R1 (Rplus R1 R1)))
*)
let int_to_real n =
   let nn=abs n in
   if nn=0
   then get coq_R0
   else
     (let s=ref (get coq_R1) in
      for i=1 to (nn-1) do s:=mkApp (get coq_Rplus,[|get coq_R1;!s|]) done;
      if n<0 then mkApp (get coq_Ropp, [|!s|]) else !s)
;;
(* -1/2 -> (Rmult (Ropp R1) (Rinv (Rplus R1 R1)))
*)
let rational_to_real x =
   let (n,d)=rational_to_fraction x in
   mkApp (get coq_Rmult,
     [|int_to_real n;mkApp(get coq_Rinv,[|int_to_real d|])|])
;;

(* preuve que 0<n*1/d
*)
let tac_zero_inf_pos gl (n,d) =
   let tacn=ref (apply (get coq_Rlt_zero_1)) in
   let tacd=ref (apply (get coq_Rlt_zero_1)) in
   for i=1 to n-1 do
       tacn:=(tclTHEN (apply (get coq_Rlt_zero_pos_plus1)) !tacn); done;
   for i=1 to d-1 do
       tacd:=(tclTHEN (apply (get coq_Rlt_zero_pos_plus1)) !tacd); done;
   (tclTHENS (apply (get coq_Rlt_mult_inv_pos)) [!tacn;!tacd])
;;

(* preuve que 0<=n*1/d
*)
let tac_zero_infeq_pos gl (n,d)=
   let tacn=ref (if n=0
                 then (apply (get coq_Rle_zero_zero))
                 else (apply (get coq_Rle_zero_1))) in
   let tacd=ref (apply (get coq_Rlt_zero_1)) in
   for i=1 to n-1 do
       tacn:=(tclTHEN (apply (get coq_Rle_zero_pos_plus1)) !tacn); done;
   for i=1 to d-1 do
       tacd:=(tclTHEN (apply (get coq_Rlt_zero_pos_plus1)) !tacd); done;
   (tclTHENS (apply (get coq_Rle_mult_inv_pos)) [!tacn;!tacd])
;;

(* preuve que 0<(-n)*(1/d) => False
*)
let tac_zero_inf_false gl (n,d) =
    if n=0 then (apply (get coq_Rnot_lt0))
    else
     (tclTHEN (apply (get coq_Rle_not_lt))
	      (tac_zero_infeq_pos gl (-n,d)))
;;

(* preuve que 0<=(-n)*(1/d) => False
*)
let tac_zero_infeq_false gl (n,d) =
     (tclTHEN (apply (get coq_Rlt_not_le_frac_opp))
	      (tac_zero_inf_pos gl (-n,d)))
;;

let create_meta () = mkMeta(Evarutil.new_meta());;

let my_cut c gl=
     let concl = pf_concl gl in
       apply_type (mkProd(Anonymous,c,concl)) [create_meta()] gl
;;

let exact = exact_check;;

let tac_use h = match h.htype with
               "Rlt" -> exact h.hname
              |"Rle" -> exact h.hname
              |"Rgt" -> (tclTHEN (apply (get coq_Rfourier_gt_to_lt))
                                (exact h.hname))
              |"Rge" -> (tclTHEN (apply (get coq_Rfourier_ge_to_le))
                                (exact h.hname))
              |"eqTLR" -> (tclTHEN (apply (get coq_Rfourier_eqLR_to_le))
                                (exact h.hname))
              |"eqTRL" -> (tclTHEN (apply (get coq_Rfourier_eqRL_to_le))
                                (exact h.hname))
              |_->assert false
;;

(*
let is_ineq (h,t) =
    match (kind_of_term t) with
	App (f,args) ->
	  (match (string_of_R_constr f) with
	       "Rlt" -> true
	     | "Rgt" -> true
	     | "Rle" -> true
	     | "Rge" -> true
(* Wrong:not in Rdefinitions: *) | "eqT" ->
                (match (string_of_R_constr args.(0)) with
			     "R" -> true
			   | _ -> false)
             | _ ->false)
      |_->false
;;
*)

let list_of_sign s = List.map (fun (x,_,z)->(x,z)) s;;

let mkAppL a =
   let l = Array.to_list a in
   mkApp(List.hd l, Array.of_list (List.tl l))
;;

(* Résolution d'inéquations linéaires dans R *)
let rec fourier gl=
    Coqlib.check_required_library ["Coq";"fourier";"Fourier"];
    let goal = strip_outer_cast (pf_concl gl) in
    let fhyp=id_of_string "new_hyp_for_fourier" in
    (* si le but est une inéquation, on introduit son contraire,
       et le but à prouver devient False *)
    try (let tac =
     match (kind_of_term goal) with
      App (f,args) ->
      (match (string_of_R_constr f) with
	     "Rlt" ->
	       (tclTHEN
	         (tclTHEN (apply (get coq_Rfourier_not_ge_lt))
			  (intro_using  fhyp))
		 fourier)
	    |"Rle" ->
	     (tclTHEN
	      (tclTHEN (apply (get coq_Rfourier_not_gt_le))
		       (intro_using  fhyp))
			fourier)
	    |"Rgt" ->
	     (tclTHEN
	      (tclTHEN (apply (get coq_Rfourier_not_le_gt))
		       (intro_using  fhyp))
	      fourier)
	    |"Rge" ->
	     (tclTHEN
	      (tclTHEN (apply (get coq_Rfourier_not_lt_ge))
		       (intro_using  fhyp))
	      fourier)
	    |_->assert false)
        |_->assert false
      in tac gl)
     with e when Errors.noncritical e ->
    (* les hypothèses *)
    let hyps = List.map (fun (h,t)-> (mkVar h,t))
                        (list_of_sign (pf_hyps gl)) in
    let lineq =ref [] in
    List.iter (fun h -> try (lineq:=(ineq1_of_constr h)@(!lineq))
		        with e when Errors.noncritical e -> ())
              hyps;
    (* lineq = les inéquations découlant des hypothèses *)
    if !lineq=[] then Util.error "No inequalities";
    let res=fourier_lineq (!lineq) in
    let tac=ref tclIDTAC in
    if res=[]
    then Util.error "fourier failed"
    (* l'algorithme de Fourier a réussi: on va en tirer une preuve Coq *)
    else (match res with
        [(cres,sres,lc)]->
    (* lc=coefficients multiplicateurs des inéquations
       qui donnent 0<cres ou 0<=cres selon sres *)
	(*print_string "Fourier's method can prove the goal...";flush stdout;*)
          let lutil=ref [] in
	  List.iter
            (fun (h,c) ->
			  if c<>r0
        		  then (lutil:=(h,c)::(!lutil)(*;
				print_rational(c);print_string " "*)))
                    (List.combine (!lineq) lc);
       (* on construit la combinaison linéaire des inéquation *)
             (match (!lutil) with
          (h1,c1)::lutil ->
	  let s=ref (h1.hstrict) in
	  let t1=ref (mkAppL [|get coq_Rmult;
	                  rational_to_real c1;
			  h1.hleft|]) in
	  let t2=ref (mkAppL [|get coq_Rmult;
	                  rational_to_real c1;
			  h1.hright|]) in
	  List.iter (fun (h,c) ->
	       s:=(!s)||(h.hstrict);
	       t1:=(mkAppL [|get coq_Rplus;
	                     !t1;
                             mkAppL [|get coq_Rmult;
                                      rational_to_real c;
			              h.hleft|] |]);
	       t2:=(mkAppL [|get coq_Rplus;
	                     !t2;
                             mkAppL [|get coq_Rmult;
                                      rational_to_real c;
			              h.hright|] |]))
               lutil;
          let ineq=mkAppL [|if (!s) then get coq_Rlt else get coq_Rle;
			      !t1;
			      !t2 |] in
	   let tc=rational_to_real cres in
       (* puis sa preuve *)
           let tac1=ref (if h1.hstrict
                         then (tclTHENS (apply (get coq_Rfourier_lt))
                                 [tac_use h1;
                                  tac_zero_inf_pos  gl
                                      (rational_to_fraction c1)])
                         else (tclTHENS (apply (get coq_Rfourier_le))
                                 [tac_use h1;
				  tac_zero_inf_pos  gl
                                      (rational_to_fraction c1)])) in
           s:=h1.hstrict;
           List.iter (fun (h,c)->
             (if (!s)
	      then (if h.hstrict
	            then tac1:=(tclTHENS (apply (get coq_Rfourier_lt_lt))
			       [!tac1;tac_use h;
			        tac_zero_inf_pos  gl
                                      (rational_to_fraction c)])
	            else tac1:=(tclTHENS (apply (get coq_Rfourier_lt_le))
			       [!tac1;tac_use h;
			        tac_zero_inf_pos  gl
                                      (rational_to_fraction c)]))
	      else (if h.hstrict
	            then tac1:=(tclTHENS (apply (get coq_Rfourier_le_lt))
			       [!tac1;tac_use h;
			        tac_zero_inf_pos  gl
                                      (rational_to_fraction c)])
	            else tac1:=(tclTHENS (apply (get coq_Rfourier_le_le))
			       [!tac1;tac_use h;
			        tac_zero_inf_pos  gl
                                      (rational_to_fraction c)])));
	     s:=(!s)||(h.hstrict))
              lutil;
           let tac2= if sres
                      then tac_zero_inf_false gl (rational_to_fraction cres)
                      else tac_zero_infeq_false gl (rational_to_fraction cres)
           in
           tac:=(tclTHENS (my_cut ineq)
                     [tclTHEN (change_in_concl None
			       (mkAppL [| get coq_not; ineq|]
				       ))
		      (tclTHEN (apply (if sres then get coq_Rnot_lt_lt
					       else get coq_Rnot_le_le))
			    (tclTHENS (Equality.replace
				       (mkAppL [|get coq_Rminus;!t2;!t1|]
					       )
				       tc)
		 	       [tac2;
                                (tclTHENS
				  (Equality.replace
				    (mkApp (get coq_Rinv,
				      [|get coq_R1|]))
				    (get coq_R1))
(* en attendant Field, ça peut aider Ring de remplacer 1/1 par 1 ... *)

      			        [tclORELSE
                                   (Ring.polynom [])
                                   tclIDTAC;
					  (tclTHEN (apply (get coq_sym_eqT))
						(apply (get coq_Rinv_1)))]

					 )
				]));
                       !tac1]);
	   tac:=(tclTHENS (cut (get coq_False))
				  [tclTHEN intro (contradiction None);
				   !tac])
      |_-> assert false) |_-> assert false
	  );
(*    ((tclTHEN !tac (tclFAIL 1 (* 1 au hasard... *))) gl) *)
      (!tac gl)
(*      ((tclABSTRACT None !tac) gl) *)

;;

(*
let fourier_tac x gl =
     fourier gl
;;

let v_fourier = add_tactic "Fourier" fourier_tac
*)

