(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)
(*                                                                      *)
(* Micromega: A reflexive tactic using the Positivstellensatz           *)
(*                                                                      *)
(*  Frédéric Besson (Irisa/Inria) 2006-2008                             *)
(*                                                                      *)
(************************************************************************)

(* We take as input a list of polynomials [p1...pn] and return an unfeasibility
   certificate polynomial. *)

type var = int



open Big_int
open Num
open Polynomial

module Mc = Micromega
module Ml2C = Mutils.CamlToCoq
module C2Ml = Mutils.CoqToCaml


open Mutils
type 'a number_spec = {
 bigint_to_number : big_int -> 'a;
 number_to_num : 'a  -> num;
 zero : 'a;
 unit : 'a;
 mult : 'a -> 'a -> 'a;
 eqb  : 'a -> 'a -> bool
}

let z_spec = {
 bigint_to_number = Ml2C.bigint ;
 number_to_num = (fun x -> Big_int (C2Ml.z_big_int x));
 zero = Mc.Z0;
 unit = Mc.Zpos Mc.XH;
 mult = Mc.Z.mul;
 eqb  = Mc.zeq_bool
}
 

let q_spec = {
 bigint_to_number = (fun x -> {Mc.qnum = Ml2C.bigint x; Mc.qden = Mc.XH});
 number_to_num = C2Ml.q_to_num;
 zero = {Mc.qnum = Mc.Z0;Mc.qden = Mc.XH};
 unit = {Mc.qnum =  (Mc.Zpos Mc.XH) ; Mc.qden = Mc.XH};
 mult = Mc.qmult;
 eqb  = Mc.qeq_bool
}

let r_spec = z_spec


let dev_form n_spec  p =
 let rec dev_form p = 
  match p with
   | Mc.PEc z ->  Poly.constant (n_spec.number_to_num z)
   | Mc.PEX v ->  Poly.variable (C2Ml.positive v)
   | Mc.PEmul(p1,p2) -> 
      let p1 = dev_form p1 in
      let p2 = dev_form p2 in
       Poly.product p1 p2 
   | Mc.PEadd(p1,p2) -> Poly.addition (dev_form p1) (dev_form p2)
   | Mc.PEopp p ->  Poly.uminus (dev_form p)
   | Mc.PEsub(p1,p2) ->  Poly.addition (dev_form p1) (Poly.uminus (dev_form p2))
   | Mc.PEpow(p,n)   ->  
      let p = dev_form p in
      let n = C2Ml.n n in
      let rec pow n = 
       if n = 0 
       then Poly.constant (n_spec.number_to_num n_spec.unit)
       else Poly.product p (pow (n-1)) in
       pow n in
  dev_form p


let monomial_to_polynomial mn = 
 Monomial.fold 
  (fun v i acc -> 
     let v = Ml2C.positive v in
     let mn = if i = 1 then Mc.PEX v else Mc.PEpow (Mc.PEX v ,Ml2C.n i) in
       if acc = Mc.PEc (Mc.Zpos Mc.XH)
       then mn 
       else Mc.PEmul(mn,acc))
   mn 
   (Mc.PEc (Mc.Zpos Mc.XH))
  


let list_to_polynomial vars l = 
 assert (List.for_all (fun x -> ceiling_num x =/ x) l);
 let var x = monomial_to_polynomial (List.nth vars x)  in
       
 let rec xtopoly p i = function
  | [] -> p
  | c::l -> if c =/  (Int 0) then xtopoly p (i+1) l 
    else let c = Mc.PEc (Ml2C.bigint (numerator c)) in
    let mn = 
     if c =  Mc.PEc (Mc.Zpos Mc.XH)
     then var i
     else Mc.PEmul (c,var i) in
    let p' = if p = Mc.PEc Mc.Z0 then mn else
      Mc.PEadd (mn, p) in
     xtopoly p' (i+1) l in
  
  xtopoly (Mc.PEc Mc.Z0) 0 l

let rec fixpoint f x =
 let y' = f x in
  if y' = x then y'
  else fixpoint f y'

let  rec_simpl_cone n_spec e = 
 let simpl_cone = 
  Mc.simpl_cone n_spec.zero n_spec.unit n_spec.mult n_spec.eqb in

 let rec rec_simpl_cone  = function
 | Mc.PsatzMulE(t1, t2) -> 
    simpl_cone  (Mc.PsatzMulE (rec_simpl_cone t1, rec_simpl_cone t2))
 | Mc.PsatzAdd(t1,t2)  -> 
    simpl_cone (Mc.PsatzAdd (rec_simpl_cone t1, rec_simpl_cone t2))
 |  x           -> simpl_cone x in
  rec_simpl_cone e
   
   
let simplify_cone n_spec c = fixpoint (rec_simpl_cone n_spec) c
 
type cone_prod = 
  Const of cone 
  | Ideal of cone *cone 
  | Mult of cone * cone 
  | Other of cone
and cone =   Mc.zWitness



let factorise_linear_cone c =
 
 let rec cone_list  c l = 
  match c with
   | Mc.PsatzAdd (x,r) -> cone_list  r (x::l)
   |  _        ->  c :: l in
  
 let factorise c1 c2 =
  match c1 , c2 with
   | Mc.PsatzMulC(x,y) , Mc.PsatzMulC(x',y') -> 
      if x = x' then Some (Mc.PsatzMulC(x, Mc.PsatzAdd(y,y'))) else None
   | Mc.PsatzMulE(x,y) , Mc.PsatzMulE(x',y') -> 
      if x = x' then Some (Mc.PsatzMulE(x, Mc.PsatzAdd(y,y'))) else None
   |  _     -> None in
  
 let rec rebuild_cone l pending  =
  match l with
   | [] -> (match pending with
      | None -> Mc.PsatzZ
      | Some p -> p
     )
   | e::l -> 
      (match pending with
       | None -> rebuild_cone l (Some e) 
       | Some p -> (match factorise p e with
          | None -> Mc.PsatzAdd(p, rebuild_cone l (Some e))
          | Some f -> rebuild_cone l (Some f) )
      ) in

  (rebuild_cone (List.sort Pervasives.compare (cone_list c [])) None)



(* The binding with Fourier might be a bit obsolete 
   -- how does it handle equalities ? *)

(* Certificates are elements of the cone such that P = 0  *)

(* To begin with, we search for certificates of the form:
   a1.p1 + ... an.pn + b1.q1 +... + bn.qn + c = 0   
   where pi >= 0 qi > 0
   ai >= 0 
   bi >= 0
   Sum bi + c >= 1
   This is a linear problem: each monomial is considered as a variable.
   Hence, we can use fourier.

   The variable c is at index 0
*)

open Mfourier

(* fold_left followed by a rev ! *)

let constrain_monomial mn l  = 
 let coeffs = List.fold_left (fun acc p -> (Poly.get mn p)::acc) [] l in
  if mn = Monomial.const
  then  
   { coeffs = Vect.from_list ((Big_int unit_big_int):: (List.rev coeffs)) ; 
     op = Eq ; 
     cst = Big_int zero_big_int  }
  else
   { coeffs = Vect.from_list ((Big_int zero_big_int):: (List.rev coeffs)) ; 
     op = Eq ; 
     cst = Big_int zero_big_int  }

    
let positivity l = 
 let rec xpositivity i l = 
  match l with
   | [] -> []
   | (_,Mc.Equal)::l -> xpositivity (i+1) l
   | (_,_)::l -> 
      {coeffs = Vect.update (i+1) (fun _ -> Int 1) Vect.null ; 
       op = Ge ; 
       cst = Int 0 }  :: (xpositivity (i+1) l)
 in
  xpositivity 0 l


let string_of_op = function
 | Mc.Strict -> "> 0" 
 | Mc.NonStrict -> ">= 0" 
 | Mc.Equal -> "= 0"
 | Mc.NonEqual -> "<> 0"



(* If the certificate includes at least one strict inequality, 
   the obtained polynomial can also be 0 *)
let build_linear_system l =

 (* Gather the monomials:  HINT add up of the polynomials ==> This does not work anymore *)
 let l' = List.map fst l in

 let module MonSet = Set.Make(Monomial) in

 let monomials = 
  List.fold_left (fun acc p -> 
                    Poly.fold (fun m _ acc -> MonSet.add m acc) p acc) 
    (MonSet.singleton Monomial.const) l'
 in  (* For each monomial, compute a constraint *)
 let s0 = 
  MonSet.fold (fun mn  res -> (constrain_monomial mn l')::res) monomials [] in
  (* I need at least something strictly positive *)
 let strict = {
  coeffs = Vect.from_list ((Big_int unit_big_int)::
                            (List.map (fun (x,y) -> 
                              match y with Mc.Strict -> 
                                  Big_int unit_big_int 
                              | _ -> Big_int zero_big_int) l));
  op = Ge ; cst = Big_int unit_big_int } in
  (* Add the positivity constraint *)
  {coeffs = Vect.from_list ([Big_int unit_big_int]) ; 
   op = Ge ; 
   cst = Big_int zero_big_int}::(strict::(positivity l)@s0)


let big_int_to_z = Ml2C.bigint
 
(* For Q, this is a pity that the certificate has been scaled 
   -- at a lower layer, certificates are using nums... *)
let make_certificate n_spec (cert,li) = 
 let bint_to_cst = n_spec.bigint_to_number in
  match cert with
   | [] -> failwith "empty_certificate"
   | e::cert' -> 
(*      let cst = match compare_big_int e zero_big_int with
       | 0 -> Mc.PsatzZ
       | 1 ->  Mc.PsatzC (bint_to_cst e) 
       | _ -> failwith "positivity error" 
      in *)
      let rec scalar_product cert l =
       match cert with
        | [] -> Mc.PsatzZ
        | c::cert -> 
	    match l with
              | [] -> failwith "make_certificate(1)"
              | i::l ->  
		  let r = scalar_product cert l in
		    match compare_big_int c  zero_big_int with
                      | -1 -> Mc.PsatzAdd (
			  Mc.PsatzMulC (Mc.Pc ( bint_to_cst c), Mc.PsatzIn (Ml2C.nat i)), 
			  r)
                      | 0  -> r
                      | _ ->  Mc.PsatzAdd (
			  Mc.PsatzMulE (Mc.PsatzC (bint_to_cst c), Mc.PsatzIn (Ml2C.nat i)),
			  r) in
        (factorise_linear_cone 
             (simplify_cone n_spec (scalar_product cert' li)))


exception Found of Monomial.t

exception Strict

let primal l = 
  let vr = ref 0 in
  let module Mmn = Map.Make(Monomial) in

  let vect_of_poly map p =
    Poly.fold (fun mn vl (map,vect) -> 
      if mn = Monomial.const 
      then (map,vect)
      else 
	let (mn,m) = try (Mmn.find mn map,map) with Not_found -> let res = (!vr, Mmn.add mn !vr map) in incr vr ; res in
	  (m,if sign_num vl = 0 then vect else (mn,vl)::vect)) p (map,[]) in
    
  let op_op = function Mc.NonStrict -> Ge |Mc.Equal -> Eq | _ -> raise Strict in

  let cmp x y = Pervasives.compare (fst x) (fst y) in

   snd (List.fold_right (fun  (p,op) (map,l) ->
      let (mp,vect) = vect_of_poly map p in  
      let cstr = {coeffs = List.sort cmp vect; op = op_op op ; cst = minus_num (Poly.get Monomial.const p)} in

	(mp,cstr::l)) l (Mmn.empty,[]))

let dual_raw_certificate (l:  (Poly.t * Mc.op1) list) = 
(*  List.iter (fun (p,op) -> Printf.fprintf stdout "%a %s 0\n" Poly.pp p (string_of_op op) ) l ; *)
    
 let sys = build_linear_system l in

   try 
   match Fourier.find_point sys with
    | Inr _ -> None
    | Inl cert ->  Some (rats_to_ints (Vect.to_list cert)) 
       (* should not use rats_to_ints *)
  with x -> 
   if debug 
   then (Printf.printf "raw certificate %s" (Printexc.to_string x);   
         flush stdout) ;
   None


let raw_certificate l = 
  try 
    let p = primal l in
      match Fourier.find_point p with
	| Inr prf -> 
	    if debug then Printf.printf "AProof : %a\n" pp_proof prf ; 
	    let cert = List.map (fun (x,n) -> x+1,n) (fst (List.hd (Proof.mk_proof p prf))) in
	      if debug then Printf.printf "CProof : %a" Vect.pp_vect cert ; 
	      Some (rats_to_ints (Vect.to_list cert))
	| Inl _   -> None
  with Strict -> 
    (* Fourier elimination should handle > *)
    dual_raw_certificate l 


let simple_linear_prover  l =
 let (lc,li) = List.split l in
  match raw_certificate lc with
   |  None -> None (* No certificate *)
   | Some cert -> Some (cert,li)
      




let linear_prover n_spec l  =
  let build_system n_spec l = 
    let li = List.combine l (interval 0 (List.length l -1)) in
    let (l1,l') = List.partition 
      (fun (x,_) -> if snd x = Mc.NonEqual then true else false) li in
      List.map 
	(fun ((x,y),i) -> match y with
             Mc.NonEqual -> failwith "cannot happen"
           |  y -> ((dev_form n_spec x, y),i)) l' in
  let l' = build_system n_spec l in 
    simple_linear_prover (*n_spec*) l' 


let linear_prover n_spec l  =
 try linear_prover n_spec l with
   x -> (print_string (Printexc.to_string x); None)

let linear_prover_with_cert spec l = 
  match linear_prover spec l with
    | None -> None
    | Some cert -> Some (make_certificate spec cert)




let make_linear_system l =
 let l' = List.map fst l in
 let monomials = List.fold_left (fun acc p -> Poly.addition p acc) 
  (Poly.constant (Int 0)) l' in
 let monomials = Poly.fold 
  (fun mn _ l -> if mn = Monomial.const then l else mn::l) monomials [] in
  (List.map (fun (c,op) -> 
   {coeffs = Vect.from_list (List.map (fun mn ->  (Poly.get mn c)) monomials) ; 
    op = op ; 
    cst = minus_num ( (Poly.get Monomial.const c))}) l
    ,monomials)


let pplus x y = Mc.PEadd(x,y)
let pmult x y = Mc.PEmul(x,y)
let pconst x = Mc.PEc x
let popp x = Mc.PEopp x
 
let debug = false
 
(* keep track of enumerated vectors *)
let rec mem p x  l = 
 match l with  [] -> false | e::l -> if p x e then true else mem p x l

let rec remove_assoc p x l = 
 match l with [] -> [] | e::l -> if p x (fst e) then
  remove_assoc p x l else e::(remove_assoc p x l) 

let eq x y = Vect.compare x y = 0

let  remove e  l  = List.fold_left (fun l x -> if eq x e then l else x::l) [] l


(* The prover is (probably) incomplete -- 
   only searching for naive cutting planes *)

let develop_constraint z_spec (e,k) = 
    match k with
      | Mc.NonStrict -> (dev_form z_spec e , Ge)
      | Mc.Equal     ->  (dev_form z_spec e , Eq)
      | _     -> assert false


let op_of_op_compat = function
  | Ge -> Mc.NonStrict
  | Eq -> Mc.Equal


let integer_vector coeffs = 
  let vars , coeffs = List.split coeffs in
    List.combine vars (List.map (fun x -> Big_int x) (rats_to_ints coeffs))

let integer_cstr {coeffs = coeffs ; op = op ; cst = cst } = 
  let vars , coeffs = List.split coeffs in
  match rats_to_ints (cst::coeffs) with
    | cst :: coeffs -> 
	{
	  coeffs = List.combine vars (List.map (fun x -> Big_int x) coeffs) ;
	  op = op ; cst = Big_int cst}
    |  _ -> assert false
	 

let pexpr_of_cstr_compat var cstr  = 
  let {coeffs = coeffs ; op = op ; cst = cst } = integer_cstr cstr in
  try 
    let expr = list_to_polynomial var (Vect.to_list coeffs) in  
    let d = Ml2C.bigint (denominator cst) in
    let n = Ml2C.bigint (numerator cst) in
      (pplus (pmult (pconst d) expr) (popp (pconst n)), op_of_op_compat op)
  with Failure _ -> failwith "pexpr_of_cstr_compat"




open Sos_types

let rec scale_term t = 
 match t with
  | Zero    -> unit_big_int , Zero
  | Const n ->  (denominator n) , Const (Big_int (numerator n))
  | Var n   -> unit_big_int , Var n
  | Inv _   -> failwith "scale_term : not implemented"
  | Opp t   -> let s, t = scale_term t in s, Opp t
  | Add(t1,t2) -> let s1,y1 = scale_term t1 and s2,y2 = scale_term t2 in
    let g = gcd_big_int s1 s2 in
    let s1' = div_big_int s1 g in
    let s2' = div_big_int s2 g in
    let e = mult_big_int g (mult_big_int s1' s2') in
     if (compare_big_int e unit_big_int) = 0
     then (unit_big_int, Add (y1,y2))
     else 	e, Add (Mul(Const (Big_int s2'), y1),
		       Mul (Const (Big_int s1'), y2))
  | Sub _ -> failwith "scale term: not implemented"
  | Mul(y,z) ->       let s1,y1 = scale_term y and s2,y2 = scale_term z in
		       mult_big_int s1 s2 , Mul (y1, y2)
  |  Pow(t,n) -> let s,t = scale_term t in
		  power_big_int_positive_int s  n , Pow(t,n)
  |   _ -> failwith "scale_term : not implemented"

let scale_term t =
 let (s,t') = scale_term t in
  s,t'


let get_index_of_ith_match f i l  =
 let rec get j res l =
  match l with
   | [] -> failwith "bad index"
   | e::l -> if f e
     then 
       (if j = i then res else get (j+1) (res+1) l )
     else get j (res+1) l in
  get 0 0 l


let rec scale_certificate pos = match pos with
 | Axiom_eq i ->  unit_big_int , Axiom_eq i
 | Axiom_le i ->  unit_big_int , Axiom_le i
 | Axiom_lt i ->   unit_big_int , Axiom_lt i
 | Monoid l   -> unit_big_int , Monoid l
 | Rational_eq n ->  (denominator n) , Rational_eq (Big_int (numerator n))
 | Rational_le n ->  (denominator n) , Rational_le (Big_int (numerator n))
 | Rational_lt n ->  (denominator n) , Rational_lt (Big_int (numerator n))
 | Square t -> let s,t' =  scale_term t in 
		mult_big_int s s , Square t'
 | Eqmul (t, y) -> let s1,y1 = scale_term t and s2,y2 = scale_certificate y in
		    mult_big_int s1 s2 , Eqmul (y1,y2)
 | Sum (y, z) -> let s1,y1 = scale_certificate y 
   and s2,y2 = scale_certificate z in
   let g = gcd_big_int s1 s2 in
   let s1' = div_big_int s1 g in
   let s2' = div_big_int s2 g in
    mult_big_int g (mult_big_int s1' s2'), 
    Sum (Product(Rational_le (Big_int s2'), y1),
	Product (Rational_le (Big_int s1'), y2))
 | Product (y, z) -> 
    let s1,y1 = scale_certificate y and s2,y2 = scale_certificate z in
     mult_big_int s1 s2 , Product (y1,y2)


open Micromega
 let rec term_to_q_expr = function
  | Const n ->  PEc (Ml2C.q n)
  | Zero   ->  PEc ( Ml2C.q (Int 0))
  | Var s   ->  PEX (Ml2C.index 
		      (int_of_string (String.sub s 1 (String.length s - 1))))
  | Mul(p1,p2) ->  PEmul(term_to_q_expr p1, term_to_q_expr p2)
  | Add(p1,p2) ->   PEadd(term_to_q_expr p1, term_to_q_expr p2)
  | Opp p ->   PEopp (term_to_q_expr p)
  | Pow(t,n) ->  PEpow (term_to_q_expr t,Ml2C.n n)
  | Sub(t1,t2) ->  PEsub (term_to_q_expr t1,  term_to_q_expr t2)
  | _ -> failwith "term_to_q_expr: not implemented"

 let term_to_q_pol e = Mc.norm_aux (Ml2C.q (Int 0)) (Ml2C.q (Int 1)) Mc.qplus  Mc.qmult Mc.qminus Mc.qopp Mc.qeq_bool (term_to_q_expr e)


 let rec product l = 
   match l with
     | [] -> Mc.PsatzZ
     | [i] -> Mc.PsatzIn (Ml2C.nat i)
     | i ::l -> Mc.PsatzMulE(Mc.PsatzIn (Ml2C.nat i), product l)


let  q_cert_of_pos  pos = 
 let rec _cert_of_pos = function
   Axiom_eq i ->  Mc.PsatzIn (Ml2C.nat i)
  | Axiom_le i ->  Mc.PsatzIn (Ml2C.nat i)
  | Axiom_lt i ->  Mc.PsatzIn (Ml2C.nat i)
  | Monoid l  -> product l
  | Rational_eq n | Rational_le n | Rational_lt n -> 
     if compare_num n (Int 0) = 0 then Mc.PsatzZ else
      Mc.PsatzC (Ml2C.q   n)
  | Square t -> Mc.PsatzSquare (term_to_q_pol  t)
  | Eqmul (t, y) -> Mc.PsatzMulC(term_to_q_pol t, _cert_of_pos y)
  | Sum (y, z) -> Mc.PsatzAdd  (_cert_of_pos y, _cert_of_pos z)
  | Product (y, z) -> Mc.PsatzMulE (_cert_of_pos y, _cert_of_pos z) in
  simplify_cone q_spec (_cert_of_pos pos)


 let rec term_to_z_expr = function
  | Const n ->  PEc (Ml2C.bigint (big_int_of_num n))
  | Zero   ->  PEc ( Z0)
  | Var s   ->  PEX (Ml2C.index 
		      (int_of_string (String.sub s 1 (String.length s - 1))))
  | Mul(p1,p2) ->  PEmul(term_to_z_expr p1, term_to_z_expr p2)
  | Add(p1,p2) ->   PEadd(term_to_z_expr p1, term_to_z_expr p2)
  | Opp p ->   PEopp (term_to_z_expr p)
  | Pow(t,n) ->  PEpow (term_to_z_expr t,Ml2C.n n)
  | Sub(t1,t2) ->  PEsub (term_to_z_expr t1,  term_to_z_expr t2)
  | _ -> failwith "term_to_z_expr: not implemented"

 let term_to_z_pol e = Mc.norm_aux (Ml2C.z 0) (Ml2C.z 1) Mc.Z.add  Mc.Z.mul Mc.Z.sub Mc.Z.opp Mc.zeq_bool (term_to_z_expr e)

let  z_cert_of_pos  pos = 
 let s,pos = (scale_certificate pos) in
 let rec _cert_of_pos = function
   Axiom_eq i ->  Mc.PsatzIn (Ml2C.nat i)
  | Axiom_le i ->  Mc.PsatzIn (Ml2C.nat i)
  | Axiom_lt i ->  Mc.PsatzIn (Ml2C.nat i)
  | Monoid l  -> product l
  | Rational_eq n | Rational_le n | Rational_lt n -> 
     if compare_num n (Int 0) = 0 then Mc.PsatzZ else
      Mc.PsatzC (Ml2C.bigint (big_int_of_num  n))
  | Square t -> Mc.PsatzSquare (term_to_z_pol  t)
  | Eqmul (t, y) -> 
      let is_unit =
	match t with
	  | Const n -> n =/ Int 1 
	  |   _     -> false in
	if is_unit
	then _cert_of_pos y
	else Mc.PsatzMulC(term_to_z_pol t, _cert_of_pos y)
  | Sum (y, z) -> Mc.PsatzAdd  (_cert_of_pos y, _cert_of_pos z)
  | Product (y, z) -> Mc.PsatzMulE (_cert_of_pos y, _cert_of_pos z) in
  simplify_cone z_spec (_cert_of_pos pos)

(** All constraints (initial or derived) have an index and have a justification i.e., proof.
    Given a constraint, all the coefficients are always integers.
*)
open Mutils
open Mfourier
open Num
open Big_int
open Polynomial

(*module Mc = Micromega*)
(*module Ml2C = Mutils.CamlToCoq
module C2Ml = Mutils.CoqToCaml
*)
let debug = false



module Env = 
struct

  type t = int list

  let id_of_hyp hyp l = 
    let rec xid_of_hyp i l = 
      match l with
      | [] -> failwith "id_of_hyp"
      | hyp'::l -> if hyp = hyp' then i else xid_of_hyp (i+1) l in
      xid_of_hyp 0 l

end


let  coq_poly_of_linpol (p,c) = 

  let pol_of_mon m =
    Monomial.fold (fun x v p -> Mc.PEmul(Mc.PEpow(Mc.PEX(Ml2C.positive x),Ml2C.n v),p)) m (Mc.PEc (Mc.Zpos Mc.XH)) in

    List.fold_left (fun acc (x,v) -> 
		      let mn = LinPoly.MonT.retrieve x in
			Mc.PEadd(Mc.PEmul(Mc.PEc (Ml2C.bigint (numerator v)), pol_of_mon mn),acc)) (Mc.PEc (Ml2C.bigint (numerator c))) p
			



let rec cmpl_prf_rule env = function
  | Hyp i | Def i -> Mc.PsatzIn (Ml2C.nat (Env.id_of_hyp i env))
  | Cst i         -> Mc.PsatzC (Ml2C.bigint i)
  | Zero          -> Mc.PsatzZ
  | MulPrf(p1,p2)      -> Mc.PsatzMulE(cmpl_prf_rule env p1, cmpl_prf_rule env p2)
  | AddPrf(p1,p2)      -> Mc.PsatzAdd(cmpl_prf_rule env p1 , cmpl_prf_rule env p2)
  | MulC(lp,p)  -> let lp = Mc.norm0 (coq_poly_of_linpol lp) in
      Mc.PsatzMulC(lp,cmpl_prf_rule env p)
  | Square lp      -> Mc.PsatzSquare (Mc.norm0 (coq_poly_of_linpol lp))
  | _                  -> failwith "Cuts should already be compiled"
      

let rec cmpl_proof env = function
  | Done ->  Mc.DoneProof
  | Step(i,p,prf) -> 
      begin
	match p with
	| CutPrf p' -> 
	    Mc.CutProof(cmpl_prf_rule env p', cmpl_proof (i::env) prf)
	|   _       -> Mc.RatProof(cmpl_prf_rule env p,cmpl_proof (i::env) prf)
      end
  | Enum(i,p1,_,p2,l) -> 
      Mc.EnumProof(cmpl_prf_rule env p1,cmpl_prf_rule env p2,List.map (cmpl_proof (i::env)) l)


let compile_proof env prf = 
  let id = 1 + proof_max_id prf in
  let _,prf = normalise_proof id prf in
    if debug then Printf.fprintf stdout "compiled proof %a\n" output_proof prf;
    cmpl_proof env prf

type prf_sys = (cstr_compat * prf_rule) list


let xlinear_prover sys = 
  match Fourier.find_point sys with
  | Inr prf -> 
      if debug then Printf.printf "AProof : %a\n" pp_proof prf ; 
      let cert = (*List.map (fun (x,n) -> x+1,n)*) (fst (List.hd (Proof.mk_proof sys prf))) in
	if debug then Printf.printf "CProof : %a" Vect.pp_vect cert ; 
	Some (rats_to_ints (Vect.to_list cert))
  | Inl _   -> None


let output_num o n = output_string o (string_of_num n)
let output_bigint o n = output_string o (string_of_big_int n)

let proof_of_farkas prf cert = 
(*  Printf.printf "\nproof_of_farkas  %a , %a \n" (pp_list output_prf_rule) prf (pp_list output_bigint) cert ;  *)
  let rec mk_farkas acc prf cert = 
    match prf, cert with
    |   _  , []   -> acc
    |  []  , _    -> failwith "proof_of_farkas : not enough hyps"
    |  p::prf,c::cert -> 
	   mk_farkas (add_proof (mul_proof c p) acc) prf cert in
  let res =  mk_farkas Zero prf cert in
    (*Printf.printf "==> %a" output_prf_rule res ; *)
    res


let linear_prover sys =
  let (sysi,prfi) = List.split sys in
    match xlinear_prover sysi with
    | None -> None
    | Some cert -> Some (proof_of_farkas prfi cert)

let linear_prover  = 
  if debug 
  then 
    fun sys -> 
      Printf.printf "<linear_prover"; flush stdout ;
      let res = linear_prover sys in
	Printf.printf ">"; flush stdout ;
	res
  else linear_prover
  



(** A single constraint can be unsat for the following reasons:
    - 0 >= c for c a negative constant
    - 0 =  c for c a non-zero constant
    - e = c  when the coeffs of e are all integers and c is rational
*)

type checksat = 
  | Tauto (* Tautology *)
  | Unsat of prf_rule (* Unsatisfiable *)
  | Cut of cstr_compat * prf_rule (* Cutting plane *)
  | Normalise of cstr_compat * prf_rule (* coefficients are relatively prime *)
      

(** [check_sat] 
    - detects constraints that are not satisfiable;
    - normalises constraints and generate cuts.
*)

let check_sat (cstr,prf) = 
  let {coeffs=coeffs ; op=op ; cst=cst} = cstr in
    match coeffs with
    | [] -> 
	if eval_op op (Int 0) cst then Tauto else Unsat prf
    | _  -> 
	let gcdi =  (gcd_list (List.map snd coeffs)) in
	let gcd = Big_int gcdi in
	  if eq_num gcd (Int 1)
	  then Normalise(cstr,prf) 
	  else
	    if sign_num (mod_num cst gcd) = 0
	    then (* We can really normalise *)
	      begin
		assert (sign_num gcd >=1 ) ;
		let cstr = {
		  coeffs = List.map (fun (x,v) -> (x, v // gcd)) coeffs; 
		  op = op ; cst = cst // gcd
		} in 
		  Normalise(cstr,Gcd(gcdi,prf))
		    (*		    Normalise(cstr,CutPrf prf)*)
	      end
	    else
	      match op with
	      | Eq -> Unsat (CutPrf prf)
	      | Ge -> 
		  let cstr = {
		    coeffs = List.map (fun (x,v) -> (x, v // gcd)) coeffs; 
		    op = op ; cst = ceiling_num (cst // gcd)
		  } in Cut(cstr,CutPrf prf)


(** Proof generating pivoting over variable v *)
let pivot v (c1,p1) (c2,p2) = 
  let {coeffs = v1 ; op = op1 ; cst = n1} = c1
  and {coeffs = v2 ; op = op2 ; cst = n2} = c2 in



  (* Could factorise gcd... *)
  let xpivot cv1 cv2 =
    (
      {coeffs = Vect.add (Vect.mul cv1 v1) (Vect.mul cv2 v2) ;
       op = Proof.add_op op1 op2 ;
       cst = n1 */ cv1 +/ n2 */ cv2 },

      AddPrf(mul_proof (numerator cv1) p1,mul_proof (numerator cv2) p2)) in
    
    match Vect.get v v1 , Vect.get v v2 with
    | None , _ | _ , None -> None
    | Some a   , Some b   ->
	if (sign_num a) * (sign_num b) = -1
	then 
	  let cv1 = abs_num b 
	  and cv2 = abs_num a  in
	    Some (xpivot cv1 cv2)
	else 
	  if op1 = Eq
          then 
	    let cv1 = minus_num (b */ (Int (sign_num a)))
	    and cv2 = abs_num a in
	      Some (xpivot cv1 cv2)
          else if op2 = Eq
	  then
	    let cv1 = abs_num b 
	    and cv2 = minus_num (a */ (Int  (sign_num b))) in
	      Some (xpivot cv1 cv2)
	  else  None (* op2 could be Eq ... this might happen *)
	    
exception FoundProof of  prf_rule

let rec simpl_sys sys = 
  List.fold_left (fun acc (c,p) -> 
		    match check_sat (c,p) with
		    | Tauto -> acc
		    | Unsat prf -> raise (FoundProof prf)
		    | Cut(c,p)  -> (c,p)::acc
		    | Normalise (c,p) -> (c,p)::acc) [] sys


(** [ext_gcd a b] is the extended Euclid algorithm.
    [ext_gcd a b = (x,y,g)] iff [ax+by=g]
    Source: http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
*)
let rec ext_gcd a b = 
  if sign_big_int b = 0 
  then (unit_big_int,zero_big_int)
  else
    let (q,r) = quomod_big_int a b in
    let (s,t) = ext_gcd b r in
      (t, sub_big_int s (mult_big_int q t))


let pp_ext_gcd a b = 
  let a' = big_int_of_int a in
  let b' = big_int_of_int b in
    
  let (x,y) = ext_gcd a' b' in
    Printf.fprintf stdout "%s * %s + %s * %s = %s\n" 
      (string_of_big_int x) (string_of_big_int a')
      (string_of_big_int y) (string_of_big_int b')
      (string_of_big_int (add_big_int (mult_big_int x a') (mult_big_int y b')))

exception Result of (int * (proof * cstr_compat))

let split_equations psys = 
  List.partition (fun (c,p) -> c.op = Eq)


let extract_coprime (c1,p1) (c2,p2) = 
  let rec exist2 vect1 vect2 = 
    match vect1 , vect2 with
    | _ , [] | [], _ -> None
    | (v1,n1)::vect1' , (v2, n2) :: vect2' -> 
	if v1 = v2
	then 
	  if compare_big_int (gcd_big_int (numerator n1) (numerator n2)) unit_big_int  = 0
	  then Some (v1,n1,n2)
	  else 
	    exist2 vect1' vect2'
	else
	  if v1 < v2 
	  then exist2 vect1' vect2
	  else exist2 vect1 vect2' in
    
    if c1.op = Eq && c2.op = Eq 
    then exist2 c1.coeffs c2.coeffs
    else None

let extract2 pred l =
  let rec xextract2 rl l = 
    match l with
    | [] -> (None,rl) (* Did not find *)
    | e::l ->
	match extract (pred e) l with
	| None,_ -> xextract2 (e::rl) l
	| Some (r,e'),l' -> Some (r,e,e'), List.rev_append rl l' in
    
    xextract2 [] l


let extract_coprime_equation psys = 
  extract2 extract_coprime psys


let apply_and_normalise f psys =
  List.fold_left (fun acc pc' -> 
		    match f pc' with
		     | None -> pc'::acc
		     | Some pc' -> 
			 match check_sat pc' with
			  | Tauto -> acc
			  | Unsat prf -> raise (FoundProof prf)
			  | Cut(c,p)  -> (c,p)::acc
			  | Normalise (c,p) -> (c,p)::acc
		 ) [] psys




let pivot_sys v pc psys = apply_and_normalise (pivot v pc) psys


let reduce_coprime psys = 
  let oeq,sys = extract_coprime_equation psys in
    match oeq with
    | None -> None (* Nothing to do *)
    | Some((v,n1,n2),(c1,p1),(c2,p2) ) -> 
	let (l1,l2) = ext_gcd (numerator n1) (numerator n2) in
	let l1' = Big_int l1 and l2' = Big_int l2 in
	let cstr = 
	  {coeffs = Vect.add (Vect.mul l1' c1.coeffs) (Vect.mul l2' c2.coeffs);
	   op = Eq ; 
	   cst = (l1' */ c1.cst) +/ (l2' */ c2.cst) 
	  } in
	let prf = add_proof (mul_proof (numerator l1') p1) (mul_proof (numerator l2') p2) in

	  Some (pivot_sys v (cstr,prf) ((c1,p1)::sys))

(** If there is an equation [eq] of the form 1.x + e = c, do a pivot over x with equation [eq] *)
let reduce_unary psys = 
  let is_unary_equation (cstr,prf) = 
    if cstr.op = Eq 
    then 
      try 
	Some (fst (List.find (fun (_,n) -> n =/ (Int 1) || n=/ (Int (-1))) cstr.coeffs))
      with Not_found -> None
    else None in

  let (oeq,sys) =  extract is_unary_equation psys in
    match oeq with
    | None -> None (* Nothing to do *)
    | Some(v,pc) -> 
	Some(pivot_sys v pc sys)

let reduce_non_lin_unary psys = 

  let is_unary_equation (cstr,prf) = 
    if cstr.op = Eq 
    then 
      try 
	let x = fst (List.find (fun (x,n) ->  (n =/ (Int 1) || n=/ (Int (-1))) && Monomial.is_var (LinPoly.MonT.retrieve x) ) cstr.coeffs) in
	let x' = LinPoly.MonT.retrieve x in
	  if List.for_all (fun (y,_) -> y = x  || snd (Monomial.div (LinPoly.MonT.retrieve y) x') = 0) cstr.coeffs 
	  then Some x
	  else None
      with Not_found -> None
    else None in


  let (oeq,sys) =   extract is_unary_equation psys in
    match oeq with
     | None -> None (* Nothing to do *)
     | Some(v,pc) -> 
	 Some(apply_and_normalise (LinPoly.pivot_eq v pc) sys)

let reduce_var_change psys = 

  let rec rel_prime vect = 
    match vect with
     | [] -> None
     | (x,v)::vect -> 
	 let v = numerator v in
	   try 
	     let (x',v') = List.find (fun (_,v') -> 
					let v' = numerator v' in 
					  eq_big_int (gcd_big_int  v v') unit_big_int) vect in
	       Some ((x,v),(x',numerator v'))
	   with Not_found -> rel_prime vect in

  let rel_prime (cstr,prf) =  if cstr.op = Eq then rel_prime cstr.coeffs else None in

  let (oeq,sys) = extract rel_prime psys in
    
    match oeq with
     | None -> None
     | Some(((x,v),(x',v')),(c,p)) -> 
	 let (l1,l2) = ext_gcd  v  v' in
	 let l1,l2 = Big_int l1 , Big_int l2 in

	 let get v vect = 
	   match Vect.get v vect with
	    | None -> Int 0
	    | Some n -> n in

	 let pivot_eq (c',p') = 
	   let {coeffs = coeffs ; op = op ; cst = cst} = c' in
	   let vx = get x coeffs in
	   let vx' = get x' coeffs in
	   let m = minus_num (vx */ l1 +/ vx' */ l2) in
	     Some ({coeffs = 
		       Vect.add (Vect.mul m c.coeffs) coeffs ; op = op ; cst = m */ c.cst +/ cst} , 
	   AddPrf(MulC(([], m),p),p')) in

	   Some (apply_and_normalise pivot_eq sys)




  let reduce_pivot psys = 
    let is_equation (cstr,prf) = 
      if cstr.op = Eq
      then
	try 
	  Some (fst (List.hd cstr.coeffs))
	with Not_found -> None
      else None in
    let (oeq,sys) = extract is_equation psys in
      match oeq with
      | None -> None (* Nothing to do *)
      | Some(v,pc) -> 
	  if debug then 
	    Printf.printf "Bad news : loss of completeness %a=%s" Vect.pp_vect (fst pc).coeffs (string_of_num (fst pc).cst); 
	  Some(pivot_sys v pc sys)





  let iterate_until_stable f x = 
    let rec iter x = 
      match f x with
      | None -> x
      | Some x' -> iter x' in
      iter x

  let rec app_funs l x = 
    match l with
    | [] -> None
    | f::fl -> 
	match f x with
	| None    -> app_funs fl x
	| Some x' -> Some x'

  let reduction_equations psys =
    iterate_until_stable (app_funs 
			    [reduce_unary ; reduce_coprime ; 
			     reduce_var_change (*; reduce_pivot*)]) psys 

  let reduction_non_lin_equations psys =
    iterate_until_stable (app_funs 
			    [reduce_non_lin_unary (*; reduce_coprime ; 
			     reduce_var_change ; reduce_pivot *)]) psys 




  (** [get_bound sys] returns upon success an interval (lb,e,ub) with proofs *)
  let get_bound sys = 
    let is_small (v,i) = 
      match Itv.range i with
      | None -> false
      | Some i -> i <=/ (Int 1) in
      
    let select_best (x1,i1) (x2,i2) = 
      if Itv.smaller_itv i1 i2
      then (x1,i1) else (x2,i2) in

    (* For lia, there are no equations => these precautions are not needed *)
    (* For nlia, there are equations => do not enumerate over equations! *)
    let all_planes sys = 
      let (eq,ineq) = List.partition (fun c -> c.op = Eq) sys in
	match eq with
	  | [] -> List.rev_map (fun c -> c.coeffs) ineq
	  | _  -> 
	      List.fold_left (fun acc c -> 
				if List.exists (fun c' -> Vect.equal c.coeffs c'.coeffs) eq
				then acc else c.coeffs ::acc) [] ineq in

    let smallest_interval = 
      List.fold_left 
	(fun acc vect ->
	   if is_small acc
	   then acc
	   else 
	     match Fourier.optimise vect sys with
	     | None -> acc
	     | Some i -> 
		 if debug then Printf.printf "Found a new bound %a" Vect.pp_vect vect ;
		 select_best (vect,i) acc)  (Vect.null, (None,None)) (all_planes sys) in
    let smallest_interval =
      match smallest_interval
      with
      | (x,(Some i, Some j))  -> Some(i,x,j)
      |   x        ->   None (* This should not be possible *)
    in
      match smallest_interval with
      | Some (lb,e,ub) -> 
	  let (lbn,lbd) = (sub_big_int (numerator lb)  unit_big_int, denominator lb) in
	  let (ubn,ubd) = (add_big_int unit_big_int (numerator ub) , denominator ub) in
	    (match
               (* x <= ub ->  x  > ub *)
               xlinear_prover   ({coeffs = Vect.mul (Big_int ubd)  e ; op = Ge ; cst = Big_int ubn} :: sys),
               (* lb <= x  -> lb > x *)
               xlinear_prover
		 ({coeffs = Vect.mul (minus_num (Big_int lbd)) e ; op = Ge ; cst = minus_num (Big_int lbn)} :: sys)
	     with
	     | Some cub , Some clb  -> Some(List.tl clb,(lb,e,ub), List.tl cub)
	     |         _            -> failwith "Interval without proof"
	    )
      | None -> None


  let check_sys sys = 
    List.for_all (fun (c,p) -> List.for_all (fun (_,n) -> sign_num n <> 0) c.coeffs) sys


  let xlia reduction_equations  sys = 

  let rec enum_proof (id:int) (sys:prf_sys) : proof option = 
    if debug then (Printf.printf "enum_proof\n" ; flush stdout) ;
    assert (check_sys sys) ; 

    let nsys,prf = List.split sys in
      match get_bound nsys with
      | None -> None (* Is the systeme really unbounded ? *)
      | Some(prf1,(lb,e,ub),prf2) -> 
	  if debug then Printf.printf "Found interval: %a in [%s;%s] -> " Vect.pp_vect e (string_of_num lb) (string_of_num ub) ; 
	  (match start_enum  id  e  (ceiling_num lb)  (floor_num ub) sys
	   with
	   | Some prfl -> 
	       Some(Enum(id,proof_of_farkas prf prf1,e, proof_of_farkas prf prf2,prfl))
	   | None -> None
	  )

  and start_enum id e clb cub sys = 
    if clb >/ cub
    then Some []
    else
      let eq = {coeffs = e ; op = Eq ; cst = clb} in
	match aux_lia (id+1) ((eq, Def id) :: sys) with
	| None -> None 
	| Some prf  -> 
	    match start_enum id e (clb +/ (Int 1)) cub sys with
	    | None -> None
	    | Some l -> Some (prf::l)

  and aux_lia (id:int)  (sys:prf_sys) : proof option  = 
    assert (check_sys sys) ; 
    if debug then Printf.printf "xlia:  %a \n" (pp_list (fun o (c,_) -> output_cstr o c)) sys ; 
    try 
      let sys = reduction_equations sys in
    if debug then 
    Printf.printf "after reduction:  %a \n" (pp_list (fun o (c,_) -> output_cstr o c)) sys ; 	    	    
	match linear_prover sys with
	 | Some prf -> Some (Step(id,prf,Done))
	 | None ->  enum_proof id sys
    with FoundProof prf -> 
      (* [reduction_equations] can find a proof *)
      Some(Step(id,prf,Done)) in

  (*  let sys' = List.map (fun (p,o) -> Mc.norm0 p , o) sys in*)
  let id  = List.length sys in
  let orpf = 
    try 
      let sys = simpl_sys sys in
	  aux_lia id sys
    with FoundProof pr -> Some(Step(id,pr,Done)) in
    match orpf with
     | None -> None
     | Some prf -> 
	 (*Printf.printf "direct proof %a\n" output_proof prf ; *)
	 let env = mapi (fun _ i -> i) sys in
	 let prf = compile_proof env prf in
	   (*try 
	     if Mc.zChecker sys' prf then Some prf else 
	     raise Certificate.BadCertificate
	     with Failure s -> (Printf.printf "%s" s ; Some prf)
	   *) Some prf
	   

  let cstr_compat_of_poly (p,o) = 
    let (v,c) = LinPoly.linpol_of_pol p in
      {coeffs = v ; op = o ; cst = minus_num c }


  let lia sys = 
    LinPoly.MonT.clear ();
    let sys = List.map (develop_constraint z_spec) sys in
    let (sys:cstr_compat list) = List.map cstr_compat_of_poly sys in
    let sys = mapi (fun c i -> (c,Hyp i)) sys in
      xlia reduction_equations sys


  let nlia sys = 
    LinPoly.MonT.clear ();
    let sys = List.map (develop_constraint z_spec) sys in
    let sys = mapi (fun c i -> (c,Hyp i)) sys in

    let is_linear =  List.for_all (fun ((p,_),_) -> Poly.is_linear p) sys in

    let module MonMap = Map.Make(Monomial) in

    let collect_square = 
      List.fold_left (fun acc ((p,_),_) -> Poly.fold 
			(fun m _ acc -> 
			   match Monomial.sqrt m with
			     | None -> acc
			     | Some s -> MonMap.add  s m acc)  p acc) MonMap.empty sys in
    let sys = MonMap.fold (fun s m acc -> 
				    let s = LinPoly.linpol_of_pol (Poly.add s (Int 1) (Poly.constant (Int 0))) in
				    let m = Poly.add m (Int 1) (Poly.constant (Int 0)) in
				    ((m, Ge), (Square s))::acc) collect_square  sys in

(*      List.iter (fun ((p,_),_) -> Printf.printf "square %a\n" Poly.pp p) gen_square*)

    let sys = 
      if is_linear then sys
      else sys @ (all_sym_pairs (fun ((c,o),p) ((c',o'),p') -> 
			    ((Poly.product c c',opMult o o'), MulPrf(p,p'))) sys) in

    let sys = List.map (fun (c,p) -> cstr_compat_of_poly c,p) sys in
    assert (check_sys sys) ; 
    xlia (if is_linear then reduction_equations else reduction_non_lin_equations) sys



(* Local Variables: *)
(* coding: utf-8 *)
(* End: *)
